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.
