Can I study the first semester in any Studienkolleg and then switch to another Studienkolleg for the second semester?
Usually it is not possible to switch between Studienkollegs. The directors of both Studienkollegs can decide about possible exceptions.
Do I need to study in the same place as where I passed my final assessment exam?
No, if you have passed your final assessment exams, you can apply to study anywhere in Germany at any university or university of applied sciences. However, in the state of North Rhine-Westphalia, there are assessment exams which are only valid for universities and universities of applied sciences in the same state.