From what I've gathered from researching this myself, cable (including internet signal) operates at the same frequencies as OTA signals. So you cannot send cable modem and OTA through a single coax through the home.
Your best bet would be to run a second coax line dedicated to the Modem.
As for an amplifier, that really depends on the strength of the OTA signals where you live. Splitting the signal to 2 TVs will drop the signal strength to each, but it may well still be strong enough without an amp. In the end, try and see if it works.