After i’m admitted to labor and delivery, am I allowed to walk?

Depends. In many cases they will get you up and walk you (with or without assistance) to help you progress in labor. If you are in premature labor & they are trying to stop it, you may be confined.There may be other reasons this would or would not be a good idea. The answer must be taylored to your individual need.
Depends. As others have stated, depends on the circumstances. If you elect for an epidural then you be in bed.
Hopefully. Unless you have a medical condition that makes it dangerous (such as very high blood pressure) you should be able to walk. This is something you should discuss with your OB provider so you are not disappointed. In general, it can make you more comfortable to be able to move and change positions. Try and find a place to give birth where this is encouraged.