Could you determine if you were moving when standing on an infinite, frictionless plane? I'm quite curious to know about this question. I don't really fully understand how inertia works so I couldn't quite put this one together in my head.
As far as I know, if you are moving in empty space, you need a reference point if you want to determine how fast you are going. If you set the reference to something moving exactly with you then your speed is 0, right?
So, imagine you are standing on an infinite hockey rink, but the surface is so perfect it has 0 friction. (Also, there is no air I guess, or any landmarks you could use for reference). Would it be possible to perform some experiment to determine if you are moving or stationary relative to the surface? (Without damaging the surface of course, no creating landmarks) If the surface had a constant and even gravity would that make a difference (i.e. could you change direction and observe different amounts of inertia depending on how fast you were going?)
Or is this situation otherwise identical to drifting in empty space with no reference points?
 A: 
If you set the reference to something moving exactly with you then your speed is 0, right?

Your velocity relative to this moving thing is zero. But you could have a non-zero  velocity relative to something else.

Would it be possible to perform some experiment to determine if you are moving or stationary relative to the surface?

All motion is relative. If every point on the surface of this rink is indistinguishable from the next, and if you are moving with a uniform velocity (no change of speed or direction, since your body can only sense changes in velocity i.e., accelerations), and since you have nothing external to gauge your motion, then no. You definitely need another reference to be able to measure your motion.

If the surface had a constant and even gravity would that make a difference (i.e. could you change direction and observe different amounts of inertia depending on how fast you were going?)

That would depend on how strong this gravitational field is. Your body can feel tidal forces, and if the field is strong enough, you will feel these forces. And changing your direction is equal to a change in velocity, or an acceleration, and your body can feel accelerations due to your inertia.

Or is this situation otherwise identical to drifting in empty space with no reference points?

Pretty much, if we consider that there are no tidal forces, no accelerations, and there is no other reference frame from which you can measure your own motion.
A: Suppose you are floating in space in your rocket and you see another rocket drifting by. Which rocket is still and which is moving? Either point of view is equally correct.
You choose a frame of reference and define motion with respect to it. A frame of reference is a coordinate system. You choose a point as the origin. Often you choose yourself. You choose directions for x, y, and z axes. You can then measure the distance along those directions to any other object. If you keep getting the same coordinates, the object isn't moving.
Generally you choose coordinates so that an object at rest stays at rest unless forces act on it. For example, you choose your rocket as the origin if the engine is off. This kind of reference frame is called inertial. Without forces, objects move at a constant velocity.
If you choose yourself as the origin, and decide the other rocket is moving, you can check the laws of physics. For example, the position of the other rocket is given by
$$\vec x = \vec x_0 + \vec vt$$
Your own position is given by the same equation, but of course $\vec x_0 = \vec 0$, and $\vec v = \vec 0$. You would find all the laws of physics work.
You can make the choice that the origin follows the other rocket, and you are moving. The laws of physics still work.
For your questions about featureless ice, ice isn't featureless. It is made of atoms. You can measure the velocity of atoms. You can treat them as the other rocket.

Gravity makes a difference, in one of two ways.
For everyday situations, you can use classical physics and describe gravity as a force. Often we deal mostly with horizontal movement. Gravity pulls us down and the ground holds us up. The two forces cancel. We can ignore them, and treat the world as an inertial frame of reference.
Or we can do physics with freely falling bodies, whose position is given by
$$\vec x = \vec x_0 + \vec vt + \frac{1}{2}\vec g t^2$$
If you take a deeper look at gravity, things change. General relativity is fundamentally a theory of gravity. It is based on the idea that you can choose any frame of reference, inertial or not. You can find ways to describe all the laws of physics that hold in any frame of reference. The consequences of this are complex and beyond the scope of this answer.
A: The ice sheet will Lorentz contract when you are moving, so it will have a minimum density (and hence gravitational field) when you are at rest.
However, there is also a graviomagentic effect caused by the mass current, which may counteract local measurements of gravity.
A detailed analysis of these effects can be found in the Supple's Paradox. (https://en.wikipedia.org/wiki/Supplee%27s_paradox), which describes the buoyancy of a relativistic submarine.
From a special relativistic view point (no gravity), there seems to be no way to determine you are moving or not...except for one thing: radiometry.
Ice emits blackbody radiation, so it would be like Earth vs. the CMB: if you measure a dipole term in the "Ice-Microwave-Background", you are moving.
Finally: if you try to imagine a frictionless surface that doesn't emit thermal radiation, you may possibly be in the same boat as Max Born trying to imagine perfectly rigid matter. Relativity just won't allow it. (See: Born Rigidity).
A: 
Would it be possible to perform some experiment to determine if you are moving or stationary relative to the surface?

If the surface were made of copper or aluminum, rather than ice, the answer would be easy. It would be yes.
The relative motion of a magnet with respect to a conductor induces a current in the conductor. That current will generate a repulsive force with the magnet. That repulsive force can be measured.
Things become a little complicated when the conductor is iron, because in addition to the force just mentioned, iron and magnets also create an attractive force. However, by manipulating the orientation of the magnet, it may be possible to calculate the repulsive force.
Things are also complicated when the surface is ice. Ice is not known as a conductor, or at least it is not a very good one. However, as small as the current that might be produced by a magnet moving relative to an ice surface might be, it is in principle measurable.
So, I would answer "yes", there are experiments that you can do that would determine whether you are in motion relative to the surface.
Now, although you can perform such an experiment, does it violate your condition that experiment proceed

Without damaging the surface of course, no creating landmarks

Whether or not inducing electrons in the surface to move is "damaging the surface" or "creating a landmark", is something for the judge to rule on.
A: We will only be able to observe relative motion in that situation , since there is no friction we will keep moving with the same velocity due to law of inertia but since motion is relative we need a point of reference , we can only define our motion in reference to another object
