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Is time simply movement? The physicist Brown said that all atoms are always moving. And all what happens simply happens because atoms move, doesn't it? So, if you could stop all atoms from moving, would there still be time?

Nice question. Is it not the case, however, that everything you say is compatible with the proposition 'Time is a dimension or framework within which things happen'? If all the atoms stopped moving then time would carry on, so to speak, but nothing would happen. Similarly, we could suggest that 'time is that which allows the measurement of movement'. If all the atoms stopped moving, there would have to be time for the statement 'they have stopped moving' to make sense.

Is it sensible to think that time is more fundamental than space, because one can just close one's eyes and relive memories, going back in time or prospectively go forward in time to predict something, without actually changing your position in space?

The thesis that time is more fundamental than space is not uncommon among philosophers -- although the significance attached to this, and the meaning of 'fundamental' varies widely. At least arguably, Aristotle, Leibniz, Kant and Heidegger, are committed to some variety of this claim. Kant's argument has some similarities to yours. All propositions about things and events must, when fully analysed, include a subordinate proposition about time (if only the location in time of the act of thought itself). But not all propositions about things and events must include a subordinate proposition about space. Kant then uses this analysis to argue further that the basic categories of all thought must be understood to be rules for the determination of time relations.