Do moving charges get affected by the magnetic field they create while moving to constitute current? If not, how can self-induction be possible? Let a bunch of charge move with a constant velocity $\mathbf v\;.$ Since, the charges are moving, they would create magnetic field $\bf B$ as it is current that produces magnetic field.
Now, would $\bf B$ affect these moving charges which themselves create $\mathbf B\;?$
My thoughts:
For electrostatics, electric field is given by the formula $$\mathbf E = \frac{q}{r^2}\hat {\mathbf {r}};$$ since $r= 0$ for the same charge that created the field, that would imply the electric field is exerting an infinite force on the charge which is not possible.
Now, if the bunch of charge is moving to constitute a steady-state current $I$ through a small-section $d\mathbf l,$ then magnetic field due to them is given by Biot-Savart Law:
$$d\mathbf B= \frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2}\;;$$ force on each charge moving through $d\mathbf l$ is given by $$\mathbf F= \mathbf v\times\frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2} \;;$$ since $d\mathbf l$ & $\hat{\mathbf r}$ are in the same direction, the numerator is zero & in the denominator $r$ is zero which would again make the force infinite. This is not possible.
However, when the current is non-steady, the moving charges produce time-dependent magnetic field which results in self-induction.
The self-induced emf caused due to change in flux of the magnetic field created by the charges does work on the charges; each charge feels force $\mathbf F= q\; \mathbf v\times \mathbf B $ where $\mathbf B$ is created by themselves.
So, that means, charges are moving & they are creating a magnetic field which again acts on the charges which created the field.
At steady state, charges are not affected by their own magnetic field whereas on the other hand, at non-steady state, the magnetic field, that those moving charges produced, imparts force on each charge leading to the production of self-emf.
But why is it so?
Fields are independent entities; they can share momentum & energy with charges as said by Timaeus.
Charges are indeed affected by their field when they read. But up to my reading, I've not found, at non-steady state of current, charges radiate. Still they are affected by the changing flux of their own created magnetic field. This produces self-emf; which means the magnetic field impart force on those moving charges which create the field.
However, why is it that moving charges are not affected by their own magnetic field when the current is at steady state but at non-steady state, charges are affected by the force from the magnetic field they create?
If charges are indeed affected by the magnetic field they create, is there any way to mathematically prove that during self-induction, the moving charges are forced by their own created magnetic field?
I'm quite confused; am I missing something here? Please help.
 A: Neither Coulomb nor Biot-Savart are correct equations for the electromagnetic field except in statics. There are time dependent generalizations, such as Jefimenko's equations.
$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}'
-\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial  \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and
$$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$
where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$
These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current. Faraday's law relates the two together.
And the next fact is also key: the EMF around a stationary wire is 100% due to the electric field. And if you have a universe with neutral objects everywhere then Jefimenko predicts the electric field is solely and 100% caused by time variation of current. And I do mean cause, as in cause and effect. The equations are causal since the charge (and change in charge) and the current (and change in current) are things in the past ($t_r\leq t$) that affect the present.
So lets review. Coulomb and Biot-Savart are incorrect when not in statics. Self induction of a stationary wire is caused by electric fields making a nonzero EMF. These electric fields are caused by time varying currents. And yes, these time varying currents also cause (and yes I do mean cause) magnetic fields.
And when people say that changing electric fields cause magnetic fields and vice versa, they are not this kind of causality where one causes the other, they are just equality since they related two things that happen at the same time. Causality relates the present (the effects) to the past (the causes). To quote Jefimenko:

neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents (emphasis added).

Now let's get to self fields. It is easy to imagine that each charge only reacts to the fields of the other charges. But do not over simplify when considering radiation reaction and other such complications.
In reality you have charges and fields. And you have to specify both. Each has energy, each has momentum, and they exchange energy and momentum with each other. Fields are not just mathematical fictions to compute forces between particles, they are real things with their own degrees of freedom, their own energy, momentum, and even stress.
Then you can find out the energy and momentum of the fields and the charges and then find out how they mutually exchange energy and momentum amongst themselves.

But as everyone said, there can be self-force only if the charges are radiating.

That is not what everyone says. Self force is much more general than radiation reaction forces. The most common case of harmonic motion had the particle gain as much energy as it loses from the inductive fields (Schott fields, that fall off faster than the radiation fields but still store energy) and only on average loses energy to the radiative fields.
Lots of people study this. Stephen Lyle wrote an entire book on just a uniformly accelerating charged particle and an entire book on self forces. Herbert Spohn wrote an entire book on the mutual coupled dynamics of electromagnetic fields and charges. And Fritz Rohrlich wrote a classic book on Classical Charged Particles. And those are just books, there are probably at least a dozen articles published every year for at least the last hundred years on the topic.
So. There are lots of self forces. Debate continues to this day about whether it is changing forces, jerk (changing acceleration), or acceleration itself that causes self forces. Or whether it's something completely different.
Keep in mind that a textbook case of a classical sized wire is usually discussing macroscopic fields, where you average over a large enough number of atoms that the fields and charge and current become smoother fields that don't jump around due to thermal effects or based on where in the lattice of the solid you are. And when someone does care about those effects, they use statistical physics and/or quantum physics. Not just pure Maxwell.
A: 
Let a bunch of charge move with a constant velocity $\mathbf v\;.$ Since, the charges are moving, they would create magnetic field $\bf B$ as it is current that produces magnetic field.

We're all familiar with the current-in-the-wire as per the picture below, but moving charges don't really "create" a magnetic field. See Jefimenko for a bit about that: "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously..." 
 
You can work this out by imagining that you're a positron, and I set you down in front of a nailed-down electron. This electron has an electromagnetic field. It doesn't have an electric field or a magnetic field, it has an electromagnetic field. As does the positron. Because of this, when I let you go, you move linearly towards the electron. Then because you only see a linear motion, you think in terms of the radial electric field as per Andrew Duffy's Physics 106 course: 
 
But what happens when I throw you past the electron? You still move towards it, but you also experience a rotational motion, which you attribute to the electron's magnetic field. But note that you didn't create the electron's magnetic field just because you moved. All that happened is that your relative motion revealed the rotational magnetic aspect of the electromagnetic interaction.     

I always thought charge cannot be affected by its own field, be it electric or magnetic.

It isn't. The electron has an electromagnetic field, and it's affected by another charged particle, or by a photon. Not by itself. 

Electric field is given by the formula $\mathbf E = \frac{q}{r^2}\hat {\mathbf {r}};$ since $r= 0$ for the same charge that created the field, that would imply the electric field is exerting an infinite force on the charge which is not possible.

Agreed. This is what Frank Close referred to in his book The Infinity Puzzle. I don't think it's much of a puzzle myself. It's quantum field theory, not quantum point-particle theory. It's the wave nature of matter, not the point-particle nature of matter. 

Now, if the bunch of charge is moving to constitute current $I$ through a small-section $d\mathbf l,$ then magnetic field due to them is given by Biot-Savart Law: $d\mathbf B= \frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2}\;;$ force on each charge moving through $d\mathbf l$ is given by $$\mathbf F= \mathbf v\times\frac{I\;d\mathbf l\times \hat{\mathbf{r}}}{r^2} \;;$$ since $d\mathbf l$ & $\hat{\mathbf r}$ are in the same direction, the numerator is zero & in the denominator $r$ is zero which would again make the force infinite. This is not possible.

Agreed. Point particles make the maths easy, but it's only an approximation, and making the maths easy shouldn't blind people to the hard scientific evidence of things like electron diffraction and the Einstein-de Haas effect. But it does. 

Conclusion: Charges cannot be affected by their own electric field & magnetic field.

Agreed. 

But then, how could there be any self-induction if moving charges cannot be affected by their own magnetic fields?

Self-inductance is "the induction of a voltage in a current-carrying wire when the current in the wire itself is changing". And that current is changing because there's some kind of interaction going on with other charged particles. Like a whole bunch of charged particles in the guise of my hand flipping a switch. 

The self-induced emf caused due to change in flux of the magnetic field created by the charges does work on the charges; each charge feels force $\mathbf F= q\; mathbf v\times \mathbf B $ where $\mathbf B$ is created by themselves. So, that means, charges are moving & they are creating a magnetic field which again acts on the charges which created the field. 

Just simplify that current to one electron, then imagine it's not moving and it's you moving instead. Nobody created a magnetic field for the electron just because they moved. Because it's got an electromagnetic field. And electromagnetic field interactions result in linear and/or rotational motion. When we only see the former we call it an electric field, when we only see the latter we call it a magnetic field, but the fields concerned are electromagnetic fields. And the simplest example of electromagnetic field interactions involves positronium. There's a linear motion such that the electron and the positron are moving towards each other and will annihilate, and  there's a rotational motion too: 

CCASA image by Manticorp/Rubber Duck, see Wikipedia 

But how can it be? That would yield an infinite force (even if not infinite, it would have to be zero since $d\mathbf l$ & $\hat{\mathbf r}$ are parallel). 

There are no infinite forces, the electron has its Compton wavelength, and it has a "spinor" nature.  

So, the million dollar question is: Do moving charges get affected by the magnetic field they create while moving to constitute current? How can it be if the formula is telling otherwise?

No they don't, because your motion relative to a charged particle doesn't create a magnetic field for it. It just results in rotational force in addition to the linear force because of the "spinor" nature of charged particles. 
 
A: Have you ever thought yourself that the magnetic dipole moments are the reason for any magnetic field? if they are aligned, they form an external field. The alignment happens by acceleration because the magnetic dipole moment is bonded to the intrinsic spin (it's really a rotation according to a really performed experiment from Einstein and de Haas). And that the statement of moving charges is a misinterpretation, the charges have to be accelerated (like in a coil). The experiment with magnetic field between two wires was performed first with switch on of the currents (an acceleration of charges).
