Questions on the notion of an inertial frame of reference Imagine an astronaut flying in an spaceship at a constant speed in completely empty space, with no acceleration whatsoever, and no outside references, just dark empty space.
Q: Can the astronaut tell at what speed is he moving? and how do you define speed in that case when you have no outside references. Can you refer to the speed relative to the empty space?
Q: Can you identify the direction of your movement? Let's say ship moving nose forward, or tail forward? Is there a sense of "direction of movement" in such an inertial frame of reference?
Q: Imaging that the speed of that spacecraft is almost the speed of light "c - 1 m/s". But no acceleration. Meaning it's an inertial frame of reference right? Will you be able to throw a ball in the direction of movement? What about the direction inverse to the movement? Does that means that there is a direction of movement in an inertial frame of reference?
 A: 
Q: Can the astronaut tell at what speed is he moving? and how do you define speed in that case when you have no outside references. Can you refer to the speed relative to the empty space?

No. This is already the case in Newtonian mechanics. Speed is always relative to some reference frame.

Q: Can you identify the direction of your movement? Let's say ship moving nose forward, or tail forward? Is there a sense of "direction of movement" in such an inertial frame of reference?

Again, movement is defined only with reference to some frame. If you are on a moving train and you start walking toward a toilet at the front of the train, you are actually experiencing train moving backwards, even though someone stationed on the ground would swear it moves forward.
In your case there is only one reference frame and that is the frame of the spaceship and everything is at rest in its own frame of reference (things are not moving relative to themselves).

Q: Imaging that the speed of that spacecraft is almost the speed of light "c - 1 m/s". But no acceleration. Meaning it's an inertial frame of reference right? Will you be able to throw a ball in the direction of movement? What about the direction inverse to the movement? Does that means that there is a direction of movement in an inertial frame of reference?

Again, the same answer applies. Since there is no acceleration, the frame of spacecraft is inertial. You claim it moves with certain speed, but relative to what? You are also moving with the speed close to the speed of light relative to some neutrino that just passed through you.
There is a relativity principle that claims, that all inertial movements are equivalent. Meaning, even though you might be moving with enormous speed relative to someone else, you will always feel like you are at rest and never observe any weird physics in the spacecraft.
A: Yes, the astronaut can easily tell the speed of his movement.  The speed (in his frame) is zero.  He can also quite easily calculate his speed in any other frame he cares to calculate in, though it's unlikely he'd find this interesting.   As for imagining that the speed is close to $c$, it is automatically close to $c$ in some frames and not in others, just like your own speed right now.
A: Your three questions are linked, in that the answer to the first gives you the answer to the second and third.
There is no such thing as an absolute velocity, or, for that matter, an absolute distance or an absolute time. Motion is relative. The astronaut in space will be motionless in her own frame of reference and be moving relative to some others.
There is no absolute direction. When you walk towards one person, you can be walking away from another.
It is meaningless to talk about moving at close to the speed of light without saying what you are moving relative to. As you sit reading this answer, you are moving at close to the speed of light relative to the most distant parts of the universe. Nothing stops you from accelerating in your own frame. It does not matter how long you accelerate for- in your own frame, you can continue to accelerate. It is analogous to approaching the horizon. Wherever you are, you can always move in the direction of the horizon, but no matter how long you travel, the horizon will continue to be out of reach.
