Can physics (ever) explain intrinsic properties of nature? I may be totally off with this quite abstract (?) question(s).
But still, here are some closely related sub-questions:


*

*Is there a list of currently "known" intrinsic properties of nature?

*How exactly is an intrinsic property of nature defined? Is it defined as "it is so, because it is so" or "it is so, because it cannot be otherwise (in our Universe)"?


For example, Wikipedia about mass as intrinsic property:

An intrinsic property is a property that an object or a thing has of
  itself, independently of other things, including its context. An
  extrinsic (or relational) property is a property that depends on a
  thing's relationship with other things. For example, mass is an
  intrinsic property of any physical object.

Is this a correct explanation of an intrinsic property? 
 A: The idea is correct. 
From my point of view, a more precise definition of intrinsic physical reality, is  a physical quantity which does not depend on the observer.
For instance, different observers can observe the same particle, and they will find the same mass for this particle. This is not true, of course, for the momentum/energy of the particle. The mass does not depend on the observer, this is an intrinsic characteristic of the particle. 
So, intrinsic physical quantities are observer-invariants.
A: Your second question may be answered so, from the anthropic principle. Things are the way they are because if it would've been any different, we wouldn't have been here to ask this question. This is only a principle and tries to explain the seeming compatibility of intrinsic properties to facilitate life and consciousness. So, the argument is, if the mass of the proton, neutron or the electron (or indeed even subatomic particles) was any different than what it is now, they wouldn't have formed atoms which formed molecules which later formed the building blocks of life all the way to the organism. 
A: I don't think my answer is appropriate for this site because I have to begin by saying "in my opinion"... :)
Physics is a mathematical science and mathematics is relational. By that I mean a quantity does not have meaning by itself, there is no Kantian thing-in-itself in mathematics. If I say $A$ and $B$ that is, by itself, meaningless, but if I say $A=B$ then I have established some mathematical relation which has meaning in the mathematical sense. This type of understanding, together with logic, forms mathematics. Physics, similarly, establishes relations between quantities with the addition that those quantities are physically measurable. They also acquire meaning through experimental observation. As physics progresses we see some quantities are deeper than others, also relations are found which render entire sets of quantities redundant (i.e., they may be viewed as following from one quantity). This is reductionism and it is a philosophical approach which served physics well. 
In other words, there are no intrinsic properties in nature, only deeper and less deep quantities. One day it is likely we will find one deep property from which everything will follow (maybe vibrational modes of a string, I don't know). 
A: If nature is supposed to encompass the whole universe, there can be no
extrinsic property of nature.
Now, regarding whether physics explains (intrinsic) properties of
nature, I think one has to be careful about the meaning of words.
Science is not supposed to explain: that is the role of religion or possibly
philosophy.  Science is supposed to describe nature, and relations
between aspects of nature that are measurable in some way, and that
remain invariant in ways to be established.
This invariance is what makes the scientific description valuable, as
the knowledge of the relations can be used to predict future evolution
of natural phenomena from initial data (more precisely from boundary
data).
As we integrate these descriptions in our thought processes, we tend
to perceive the more accepted ones as natural truth from which other
truth can be explained by the relations that were identified. But this
is a dangerous bias, that can also prevent us from making deeper
observation that would contradict it.
Science organizes knowledge of nature, it does not explain it. That is
true of physics, but also or other sciences.
Of course, we commonly say that this law of physics explains such and
such phenomenon. But that is colloquial language to indicate the
existence of a relation that given some observed hypothesis will lead
to the observed conclusion. Because of the invariance.
Now, what is a property of nature? What exists? I am not sure the
question has meaning. Probably anything that can be observed,
preferably measured, and put in relation with other such things.
Since they are in relation, it may be that given the density of
relations (say "equations") some of these concepts/things/properties (say
variables) are redundant, as they can be deduced from others.
But we do not have to agree on which are. It is often in the eyes of
the beholder, or the instruments of the scientist.  And it is also
often a matter of scale, or of precision, or a matter of simplicity. Some views may be simpler in some cases, and more complex in others.
People dismiss old physical theories too easily. Geocentricity of the
universe is a perfectly good theory when you know its domain of
application, such as running the ocean. It certainly does not work as
well to explain peculiarities of Mercury. So, what is nature ?
However, if physics does not explain, mathematics can provide some
explanation. Mathematics is often considered as outside, beyond
nature, though that may also be questionned in some precise sense.
Physical phenomena are describable by mathematical relations, that
have to obey the laws of the corresponding mathematical theories, if
the description is supposed to be adequate.  Then the consistency of
mathematics may impose other relations that must be observable, if the
theory is not to be questionned. So some natural phenomena, or some
properties of natural elements (as much as make sense, as said above)
may be explained by mathematical consistency, assuming that other
phenomena are considered part of reality as described.
