Yes, movement is undetectable unless you have something to track your movement against. In fact, there’s no way to define “staying still” that’s true for everyone in the universe.
You can only feel acceleration, not constant speed movement.
2x the speed of light is impossible. As things move faster, time slows down so they can’t make as big of a change to their movement speed using the same amount of energy. Instead, lengths contract so your destination doesn’t appear to be as far as it would have been at the start of your journey.