1. Can I get arrested when I am crossing terrain?
2. Faction A has a facility on a planet controlled by faction B. Does the rooms in the facility considered as the territory of faction B but not faction A?
1. I'd say so, as far as I know you appear on as traveling on the square you started from, so people can still see you till you finish travel and appear on the next square.
2. I believe it would be faction B, that owns the planet. It doesn't list facilities as exception to territory, only capital ships.