Inertial and non-inertial frames in classical mechanics Does the inertia of a mechanical system depend on the choice of coordinate systems? For example, is there a mechanical system that is non-inertial in Cartesian, but that is inertial in spherical?
 A: To answer your question we first need to understand how a reference frame is defined as inertial in classical mechanics (CM). 
The proper way to define an inertial frame in CM is not at all as simple as it might seem. Note that the struggle behind the definition of inertial frame is only completely solved in general relativity, by a more solid ri-definition of the concept of inertial reference frame, but this is not what we want to talk about here, so from now on let's stick to CM:
The naive way to define an inertial reference frame in CM is the following:

An inertial reference frame is a frame that is non accelerating

At a first glance this seems to be a good definition, but we have to keep in mind that we can only measure velocity and acceleration with respect to something else. If you see an object accelerating maybe that object is indeed accelerating, or maybe the object is stationary and you are accelerating. So you can see that in classical mechanics concepts of acceleration, and motion in general, depend on some sort of external reference that is non moving! So the definition has to be something like:

An inertial frame is a frame that is not accelerating with respect to a stationary frame

But the concept of a stationary frame is exactly what we are trying to define here! Stationary frame in classical mechanics is a synonym of inertial frame! So we can see that the second attempt really is stating:

An inertial frame is a frame that is not accelerating with respect to an inertial frame

That is completely inadeguate since we are using in the definition the concept that we are trying to define!
So what now? Well we can try to use the phenomenon of fictitious forces that an accelerating frame has to experience:

An inertial frame is a frame with no fictitious forces

But this doesn't work either! How are we suppose to know which force is fictitious and which is not? Is the same problem as before, but putted in a different perspective! Note also that gravity, that seems so real as a force, turns out to be best described as a fictitious force in general relativity! So this struggle is really important and consequential. Anyway this is clearly not a good definition, so what now?
The best definition for inertial frame in CM turns out to be the following:

An inertial frame is a frame far away from any other object

This seems a really strange definition! Keep in mind that what we are trying to define here is a frame that is not accelerating. But if you think about it this definition makes perfect sense! This definition is based on the assumption that: forces become weaker at long distances, meaning that if an object is really far away from you it cannot apply a noticeable amount of force to you. This is of course an assumption, a postulate, is not a proven fact, but our universe seem to behave like so, as experimental evidence show. So using this assumption we can say that if we put our reference frame really far away from everything else then nothing will exert a force on it, and so it will be non accelerating (in fact $F=ma$) and so it will be an inertial reference frame!
From this definition of reference frame we can see that the structure of the coordinate system does not play any role in determining if it is inertial or not! So the answer to your question is no, the inertiality of a reference frame does not depend of the choice of coordinate system (cartesian, polar, ecc.) And this simply follows directly from our definition.
But of course our CM definition of inertial frame is far from perfect: it depends on the mentioned postulate and also it depends on the vague concept of "far away". Sadly this is the best we can do in CM, and for any pratical application we take something that is non moving with respect to what we care abaut in our problem and call it a day. But worry not! As I mentionend in general relativity all this mess will be solved.
A: No, there is not. A reference system  is an Euclidean  rest space and it does not matter which coordinate system you use therein.  The fact that a reference system is inertial or not has no relation with the coordinate system you adopt to describe the position of material point in that rest space.
A reference system is a kinematical notion a coordinate system is a geometrical notion in a given reference system.
A: No, if a frame is inertial with one of the systems, it is inertial for the others.
But one advantage of choosing the cartesian coordinates for an inertial frame is that any inertial movement can be easily recognized, because it keeps the proportionality between the position vector components. Along the movement through points $P_0, P_1$ and $P_2$:
$$\frac{x_0-x_1}{y_0 - y_1} = \frac{x_1-x_2}{y_1 - y_2}$$
The same for $x , z$ and $y , z$. It doesn't happen for curvilinear coordinates.
A: This is actually some kind of argument that Newton had for his 'absolute frame of reference' if I'm not mistaken, 'called the bucket argument', see here. I think Mach's and Einstein's view of this was then that you should consider the viewpoint from 'stars at infinity' i.e the rotating frame isn't a 'good' reference frame. But the wikipedia article probably does a better job at explaining it.
