The two existing answers are good, with many good technical points. While respecting these answers I will try to give another answer that I wonder may perhaps be more satisfying to you.
The fundamental crux of your question comes when we look at your points:
- you give a good experimental definition of movement within your room
- you give another definition of movement within the solar system
- you allude towards the possibility of a future more fundamental definition
They can all give different kinds of answers. So the answer is clearly: it changes depending on what you mean by movement. There can be no answer at all until it is decided what precisely we mean. As physics research progresses, new ideas of movement are proposed. Currently there is no evidence to suggest that spatial dimensions are one way, and in fact all the evidence points to spatial dimensions being two way. However, maybe as a silver lining for your suggestion, the newest theory of space (relativity) already includes a way for this to be possible, because there is a maximum speed we can travel, the speed of light. So if the 'background movement' was more than the speed of light, we would indeed only be able to travel one way. So that means the discovery of 'one way only movement' would not even necessarily require us to rewrite our existing science from scratch.
Some comments have been made that your proposal that there exists an absolute kind of motion and absolutely fixed points in the universe is wrong. As far as I am aware, all current scientific evidence suggests these comments are correct in that there is no way to detect "absolute" motion. This is a kind of physical proof. However, I think your question suggests towards a more strict kind of proof. What we have not discovered is any evidence that there can be no possible deeper theory with absolute motion. Indeed I would consider it harder to go from a theory of fixed positions to a relative one than vice versa, and I would even venture to say there can never be an absolute proof of a lack of relative motion. So there is always the possibility we could discover 'one way spatial dimensions' in future, even if we have no reason to expect it now.
As a bonus, what would it look like if spatial dimensions really were 'one way' and how could we tell? We would have to be able to measure something, some kind of physical property or 'constant' (not the best word in this case!) that was constantly changing. But, it would change at a different speed if you travelled one way compared to the other way. That would show we were travelling relative to something else. But no matter how fast you travel, you would not be able to stop it changing completely. That would show however fast you are going, you would always be going one way. What kind of 'constant'? It could be something like the strength of the electric field, or maybe some physics we haven't discovered yet, perhaps a property of dark matter. But just to say again, nothing like this has been discovered yet.
- To get a deep and philosophical answer to this question, you need to deeply understand what you mean philosophically by "move", "the same point" and "back", as others pointed out.
- All the physical evidence currently suggests that it is possible to move back and forth, using what most professional scientists think is the most helpful yet fundamental definition of movement. However, this issue can never be absolutely proved one way or the other - and that's part of what makes physics so exciting.