Yes, you can split the coax as needed and you can put the router on any coax tap.
I don't believe FiOS provides any TV channels when you don't subscribe to TV.
The difference is cable sends the same TV channel package to every house with access control managed by the set-tops. Unencrypted channels are available to all as there is no easy way to block them.
FiOS, on the other hand, generates the coax signal in the ONT. It is trivial for Verizon to enable and disable TV service by adjusting the ONT.
If you want local channels and don't want to pay for them you should consider an antenna.
Good Luck.