When an electron changes its spin, or any other intrinsic property, is it still the same electron? I am not asking why an intrinsic property, like spin can have more then a single value. I understand particles (electrons) can come to existence with either up or down spin. I am asking why it can change while the particle exists. 
Electrons are defined in the SM as elementary particles, and its intrinsic properties include both EM charge and spin.


The electron is a subatomic particle, symbol 
    e−
     or 
    β−
    , whose electric charge is negative one elementary charge.
    Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ.


The EM charge of the electron is defined as -1e, and the spin as 1/2.


Electrons have an electric charge of −1.602×10^−19 coulombs,[66] which is used as a standard unit of charge for subatomic particles, and is also called the elementary charge.
    The electron has an intrinsic angular momentum or spin of 
    1
    /
    2
    .[66] This property is usually stated by referring to the electron as a spin-
    1
    /
    2
     particle.


https://en.wikipedia.org/wiki/Electron


In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.[1][2]
    Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower.
    In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though this value may be equal to zero).


https://en.wikipedia.org/wiki/Spin_(physics)


he spin transition is an example of transition between two electronic states in molecular chemistry. The ability of an electron to transit from a stable to another stable (or metastable) electronic state in a reversible and detectable fashion, makes these molecular systems appealing in the field of molecular electronics.


https://en.wikipedia.org/wiki/Spin_transition
So basically an electron can change its spin from up to down or vica versa, thought it is an intrinsic property.
Electrons EM charge cannot change.


In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject.


So both EM charge and spin are intrinsic properties of electrons. Though, electrons are coming to existence with a certain EM charge and spin. Still, EM charge is unchanged as long as the electron exists, but spin can change.
I do understand that electrons can have intrinsic properties, that can have either a single value, or a set of values. I do understand that some elecrons come into existence with EM charge and spin up. Some electrons come into existence with EM charge and spin down.
What I do not understand, is, how can spin change while the electron still exists, whereas EM charge cannot, though, both are intrinsic properties.
Do we know that when an electron undergoes spin flip (spin transition), that the electron that had originally spin up is the same quantum system that after the spin transition has spin down?
Can it be that the electron before with spin up ceases to exist (vacuum fluctuation), and then another electron is coming into existence with spin down?
Why do we say that the electron that had spin up (which is an intrinsic property) is the same quantum system as the electron that has later on (after spin flip) spin down?
After the big bang, at the baryion asymmetry, some electrons came into existence with spin up and some with spin down. Do we call these the same electrons?
Is spin the only intrinsic property of the electron that can change (like helicity)?
Question:


*

*How can an intrinsic property of an electron change (spin flip)?

*Are there any intrinsic properties (of elementary particles), that do have multiple values available, but still can't change?
 A: It doesn't matter.
Suppose two electrons approach each other, exchange a photon, and leave with different spins. Are these "the same electrons" as before? This question doesn't have a well-defined answer. You started with some state of the electron quantum field and now have a different one; whether some parts of it are the "same" as before are really up to how you define the word "same". Absolutely nothing within the theory itself cares about this distinction. 
When people talk about physics to other people, they use words in order to communicate effectively. If you took a hardline stance where any change whatsoever produced a "different" electron, then it would be very difficult to talk about low-energy physics. For example, you couldn't say that one atom transferred an electron to another, because it wouldn't be the "same" electron anymore. But if you said that electron identity was always persistent, it would be difficult to talk about very high-energy physics, where electrons are freely created and destroyed. So the word "same" may be used differently in different contexts, but it doesn't actually matter. The word is a tool to describe the theory, not the theory itself.
As a general comment: you've asked a lot of questions about how words are used in physics, where you take various quotes from across this site out of context and point out that they use words slightly differently. While I appreciate that you're doing this carefully, it's not effective by itself -- it's better to learn the mathematical theory that these words are about. Mathematics is just another language, but it's a very precise one, and that precision is just what you need when studying something as difficult as quantum mechanics.

Another question, which I think you implied in your (many) questions, is: under what circumstances are excitations related by changes in intrinsic properties called the same particle? Spin up and spin down electrons are related by rotations in physical space. But protons and neutrons can be thought of as excitations of the "nucleon" field, which are related by rotations in "isospin space". That is, a proton is just an "isospin up nucleon" and the neutron is "isospin down", and the two can interconvert by emitting leptons. So why do we give them different names?
Again, at the level of the theory, there's no actual difference. You can package up the proton and neutron fields into a nucleon field, which is as simple as defining $\Psi(x) = (p(x), n(x))$, but the physical content of the theory doesn't change. Whether we think of $\Psi$ as describing one kind of particle or two depends on the context. It may be useful to work in terms of $\Psi$ when doing high-energy hadron physics, but it's useful to work in terms of $p$ and $n$ when doing nuclear physics, where the difference between them is important. 
It always comes down to what is useful in the particular problem you're studying, which can be influenced by which symmetries are broken, what perturbations apply, what is approximately conserved by the dynamics, and so on. It's just a name, anyway.
A: What people mean when they say that spin is an intrinsic property is that spin represents an internal state of the particle that exists independently of its position and motion in space. However, the value* of that internal state can and does change, and when that happens that doesn’t mean the electron can be meaningfully said to have been replaced by a “different” electron, any more than an electron that changed its position in space would be thought of as a “new” or “different” electron. We just say that the electron moved.
Similarly, there is nothing strange or inconsistent about thinking that the spin of the electron changed, and there is no need to explain the strangeness away by saying the electron has been replaced by ”another” electron. A change of spin is a completely reasonable thing to imagine, once one has overcome the small hurdle of understanding what it means for spin to be “intrinsic”. It is not the particular direction in space of the spin that is intrinsic, rather, what is intrinsic is the set of labels that spin can assume (that is, the vector space - $\mathbb{C}^2$ in the case of the electron - where spin “lives”) along with the precise rules that govern how the spin internal state evolves and interacts with position and other parameters of the quantum system.
* Another subtle issue here is that one usually cannot consistently talk about the spin having a value in the sense of a particular direction in space that the spin vector is “pointing”. This is the difficulty alluded to in @EmilioPisanty’s answer, having to do with the fact that the three coordinates of the spin operator-valued vector do not commute, which means they cannot simultaneously be thought of as having well-defined values. This issue is tangential to my remarks above, but still important to mention, as it illustrates another way in which the words that physicists use to talk about ideas in physics fail to communicate nuances of meaning that can only properly be conveyed using precise mathematical language. As @knzhou says, in order to properly understand what spin is, there is no susbstitute to learning the mathematics behind it.
A: The intrinsic angular momentum is:
$$ ||\vec J|| = \hbar \sqrt{j(j+1)} = \hbar \frac{\sqrt 3} 2 $$
and that never changes. The projection onto an axis can change, and it has eigenvalues:
$$ j\hbar = \frac 1 2 \hbar$$
Moreover, the projection can change just by changing the coordinates (say, use the $x$-basis), or by dynamics.
Regarding how we get from $|\uparrow\rangle$ to $|\downarrow\rangle$, I take the view that we know the initial and final states (in the free particle approximation), and that every possible path contributes to the transition.
Regarding the electron identity, I am not sure there is an answer (meaning it could be a classical question). If I have a stationary electron at $\vec x_0$, I can think of it as one particle that is an excitation of the electron field at:
$$ \psi(t, \vec x_0)$$
But to a moving observer it is not stationary, and I need to Lorentz transformation $t\rightarrow t'$ and $\vec x \rightarrow \vec x'$, so it is one particle, but its identity involves different points in that view of the electron field, $\psi'$. When you realize that mass is not "stuff" (a classical view), but just a coupling to the Higgs field that leads to non-zero frequency at zero momentum, then the best you can do is say its not stuff in the classical sense. It's an excitation of the electron field, and there are conserved quantities.
That view goes well with the indistinguishable particle problem in QED scattering: if there are 2 electrons in the final state, it's not that they are identical, it's that they are indistinguishable, meaning they don't really have an identity as different particles, and you have to consider both (or all) paths that lead to the final observed state of the electron field.
A: Spin is a complicated quantity in quantum mechanics. If you want to really understand it, there is absolutely no substitute for a complete reading of a full-grown textbook. (That means: Cohen-Tannoudji, Shankar, Sakurai, or equivalent level. Introductory textbooks like Griffiths are OK as an on-ramp, but not the full deal.)
Spin is complicated because it is 


*

*an operator quantity, i.e. a quantity that need not have a well-defined value;

*a vector quantity, i.e. a quantity with three independent compontents; and moreover

*a vector operator whose components are incompatible (i.e. do not commute) with each other, which means that if one component of the spin has a well-defined value, then the other two will not.


This means that the spin comes with three components, $\hat{S}_x$, $\hat{S}_y$ and $\hat{S}_z$, but only one of the three can have a well-defined value at any given time.* However, that said, there is one more relevant quantity, which is the total spin, i.e. the combination
$$
\hat{S}^2 = \hat{S}^2_x + \hat{S}^2_y + \hat{S}^2_z,
$$
which commutes with all of the individual components, and that means that the most complete set of information you can get about a system with angular momentum in three dimensions is the total spin, $S^2$, and one of the components (traditionally taken to be $S_z$, but it is crucial to emphasize that this can be along any direction you might care to choose).
Moreover, because of technical reasons to do with quantization, the possible values of these components are restricted: the total spin can only take values of the form $S^2 = \hbar^2 s(s+1)$, for $s\in \tfrac12 \mathbb N = \{0,\frac12,1,\frac32,2,\ldots\}$ a nonnegative integer or half-integer, and the total spin projection can only take the values $S_z = -\hbar s, -\hbar (s-1), \ldots, \hbar (s-1), \hbar s$. When we say that a given system "has spin $s$", what we really mean is that it has total spin $S^2 = \hbar^2 s(s+1)$.
For electrons, these two quantities play very different roles.


*

*The total spin is intrinsic. All electrons have total spin quantum number $s=1/2$, which means that they have total spin $S^2 = \frac34\hbar^2 $, and nothing you can do to an electron will change this.

*The spin projection, $S_z$, on the other hand, is not intrinsic, and it basically tells you which direction (within the bounds of the quantization of angular momentum) the spin is pointing.


When you do things like spin flips with an electron, you're changing the latter, not the former.
$\ $
 * With one exception when they're all zero, with total spin zero.
A: None of the answers here really get at the point directly enough - which is that this question, itself, is based on a misunderstood motivation.
In particular, there is a basic misunderstanding here by the original poster of what constitutes the "intrinsic spin" of the electron, such that what is happening in a spin-up/down flip is being viewed as some form of change to that intrinsic property, and then trying to ask as to how that makes sense and is not a contradiction.
However, this is wrong. You see, spin, which is really just a kind of angular momentum, is a vector quantity: a mathematical object which allows us to encode in a conveniently-manipulatable package both an actual amount of something, or magnitude, plus an associated notion of direction.
Now why is angular momentum a vector quantity? This is because it is a kind of measure of the rotation of an object. To specify how something is rotating, you need two pieces of information: one is how fast it's rotating, while the other is the axis about which it rotates. Think about the Earth - it is rotating through an axis directed through Antarctica to the Arctic Ocean, but it need not be that way. You could imagine it rotating instead through an axis between the central United States and the Indian Ocean, instead, or between (a suitable spot in) China and Argentina. Moreover, it is rotating with a given speed: one turn every 86.164 ks (not the 86.4 ks [24 h] that makes what we usually call "a day" - that's a topic in its own right). The magnitude is related to the speed, while the direction of the angular momentum vector sets the rotation axis.
Now with electrons, of course, this is quantum mechanics - and the classical concept of rotation, taken literally, doesn't work so much anymore: for one, you cannot reasonably assign a "speed" of some kind of internal motion to it, but you can still assign it a magnitude of angular momentum, and this amount of angular momentum is fixed for every electron. What instead changes in a "spin-flip" is the direction of angular momentum. Effectively, if you want to still talk of "rotational speed" even though it has no clear referent in this realm anymore, you can say it keeps to the same "speed", but its axis of rotation shifts. The latter is not intrinsic, but only the magnitude.
And the reason for this shift is an interplay of forces. In fact, this same thing happens in classical mechanical situations as well: if you have an object that is rotating, like the Earth, you can cause, with the right external forces, this axis to realign itself. The same goes here in the case of the atom, only the relevant forces are magnetic forces between the electron and nucleus.
