I am pretty sure this is not possible with FiOS, your ONT on your house is set to the number of TV's that will get the signal and if you start splitting that signal you will start having problems with your picture. You must purchase a box for each TV with the new digital change-over to get the service on each TV.
The answer is yes you can split the coax between the TV and router as long as the RF signal on that coax is strong enough. Since you do not have a meter to check the signal you will have to try the trial and error method. If you try it and the TV picture gets worse or is missing channels then your answer is no.