Is it normal for a doorbell transformer to be warm and make noise?
Transformers are always going to produce some heat. It's a part of the step-down process. It should only be warm to the touch, however. If it burns you, there's something seriously wrong.
Some noise is not unusual either. Go stand near a power company transformer and you should hear some noise as well. If the transformer is cheap, it might make more noise than a more expensive model (my experience only, yours may differ), but they generally aren't silent. #1 would worry me more than this