Inertial frames of reference I'm struggling with the notion of an inertial frame of reference. I suspect my difficulty lies with the difference between Newtonian and relativistic inertial frames, but I can't see it. 
I've read that Newton's laws apply in any non accelerating frame of reference, which are called inertial frames. So, if I play pool on a train moving with uniform velocity, the balls behave in the same way as if I were playing pool in a pool hall. So the train is (to a good approximation, ignoring forces caused by the Earth's rotation and movement around the Sun etc) an inertial frame. 
I've also read that in an inertial frame, Newton's first law is satisfied. So, if I slide a rock on a sheet of ice and I could somehow eliminate the frictional forces between the rock and the ice, the rock would carry on sliding for ever, as predicted by Newton's first law. 
Question 1 - is the  sheet of ice also therefore a good approximation to an inertial frame?
Question 2 - are these two definitions of inertial frames saying the same thing in different ways?
Both these examples occur in gravitation fields, which doesn't seem to matter as both the train and the ice sheet are presumably good approximations to inertial frames.
Question 3 - is gravity irrelevant when defining these two inertial frames?
In special relativity, I've read (Foster and Nightingale) that an inertial frame is also one where Newton's first law holds. But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity.
Question 4 - how come you can use the same definition of an inertial frame (satisfied Newton's first law) but in the case of special relativity, the train and the ice sheet aren't inertial frames. Is this to do with not being able to synchronize clocks in a gravitational field?
In general relativity, I've read (Schutz) that a freely falling frame is (on Earth) the only possible (and local) inertial frame. So again the first two examples would not be inertial frames in general relativity.
Question 5 - am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?
Question 6 - have I missed anything else that might be useful?
Many thanks
Edit. On reflection, am I right in thinking that because Newtonian mechanics assumes universal, absolute time we don't need to worry about synchronizing clocks in a Newtonian inertial frame. Therefore we don't need to worry about gravity in a Newtonian inertial frame, because in such a frame gravity does not affect time.
This is not the case in spacetime, as here gravity does affect time and the only way to have synchronized clocks in an inertial frame in spacetime is:
1. have no gravitational field, or
2. use a local, freely falling frame.
Am I on the right track here? 
 A: "But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity." SR simply doesn't apply. That doesn't mean that these examples are noninertial frames according to SR; it just means that SR can't discuss the situation at all. (Of course, the gravitational field might be small enough to ignore for the purposes of analyzing a given situation using SR.)
The definition of an inertial frame in SR is essentially the same as in Newtonian mechanics.
"am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?" Yes in GR. No in SR.
A: In Newtonian mechanics, special relativity, general relativity everywhere the definition of inertial frame is the same.

If Newton's laws of motion are valid in a reference frame then you call the frame an inertial frame. Any frame which is moving with constant velocity with respect to this frame is also an inertial frame since Newton's laws will be equally valid in those frames as well.

The earth is only an approximate inertial frame, so is a friction less ice sheet as you yourself have said. Newton's laws are only approximately valid in those frames.
In Special relativity also, a frame is an inertial frame if Newton's laws of motion hold good. These are frames which you get only if gravity is absent.
In the presence of an uniform gravitational field, any freely falling reference frame is an inertial reference frame since again Newton's laws hold good only in those freely falling frames, for example a freely falling elevator in an uniform gravitational field. If an observer in that elevator carry out some experiment she can prove that Newton's laws are valid within the elevator.
This is due to the equivalence principle of general relativity. In a realist case, one would have to go to infinitesimally small region for the validity of the equivalence principle and in those infinitesimal elevators Newton's laws will hold.
A: Answer 1 - The sheet of ice can be considered a good example of an inertial frame, but the nearly frictionless surface of the ice isn't necessary to make it an inertial frame.  As you said:

I've also read that in an inertial frame, Newton's first law is satisfied.  

Which is true.  The ideal frictionless ice surface would therefore be a good place to test and see if Newton's first law holds, which it does for the rock you described.  But if you were to slide a rock on a surface with friction, that doesn't mean that Newton's first law is violated, because now there is a force.  The ground could still be considered in an inertial reference frame (minus the gravity, which we will talk about later) but the rock, if it considers itself as a frame of reference, will not be in an inertial reference frame because objects at motion relative to it are not staying in motion despite no force existing on those objects (except the force it has with the ground.  This force would not apply to a passing tree, however).  However, if it uses an outside point as a reference frame then it can measure its speed and acceleration relative to that inertial reference frame, using Newton's second law and the force it has with the ground.  I hope that made sense and answered your question well.
Answer 2 - The rock example, I hope, shows that these two definitions of an inertial frame are indeed related.  If I am accelerating relative to something else(in violation of the first definition you had of inertial frames), then that object will seem to be accelerating without an outside force acting on it (in violation of the the second definition you had).  So, these two definitions are very much related.
Answer 3 - Speaking classically, gravity doesn't matter when talking about these reference frames, because in both of them there is a normal force balancing the gravitational force, so the net force (and therefore net acceleration) are kept at zero, so the frame of reference is not accelerating and therefore is inertial.  Gravity does matter when speaking about relativity though (next few questions).
Answer 4 - As others have talked about in their answers, SPECIAL relativity doesn't deal with accelerating reference frames at all.  Special relativity is the non-accelerating SPECIAL case of general relativity (hence SPECIAL and GENERAL relativity).  I am not sure if the case where an object is held without acceleration by both gravity and a normal force constitutes an inertial frame in relativity the same as in classical physics, hopefully someone else can say more.
Answer 5 - Not sure (and it would be GR we want to talk about, not SR).  Anyone can answer this question better?
A: Observation from any inertal frame of reference would be eqivalent to a nonrelative location as far as the speed of light is concerned.  The photon presents itself to the observer independent of the source of motion from which it comes from; i.e. any frame of inertial reference.
