a) Yes.
b) You will find a wealth of information at uspto.gov. It is advisable to engage an attorney to assist in the application process.
c) Will A be forced to cease the TM.com domain? - Not as a matter of course. How such a dispute would pan out depends on the circumstances of the case. For example, the term "Scrabble" is owned by different parties in different countries. If you go to
www.scrabble.com, you are first asked to indicate where you are. Things of this sort are managed in the same way that any kind of dispute is managed - whatever the parties can reach agreement to or are told to do by a competent authority, if they can't agree.