Should I be worried about getting a bloody nose a week after my tonsillectomy?

Not really. It should not mean anything more whether or not you had a tonsillectomy. Treat with local pressure and intranasal moisture. Intranasal Bacitracin can help especially during the cold dry weather.
Not usually. Not from nose itself. But if you had adenoidectomy with tonsillectomy, some risk for nose bleeds. If you are actively bleeding from nose or mouth after tonsillectomy, you should go to your surgeon's office or nearest er for evaluation right away.