FiOS (and increasingly many cable companies) run most local over-the-air channels you could receive in clear-QAM over the cable. This means if your TV has a QAM tuner in it (most newer models do) you would be able to receive those channels. However all the other channels are encrypted QAM (both traditional cable channels as well as subscription services). To receive those channels, your TV needs to either be outfitted with a cable-Card or utilize a set-top BOX to be able to decode the signals.
Cable cards run $3.95/month from Verizon. There's also an SD "digital adapter" for $3.95/month which can decode all signals but which doesn't show the STB guide or have access to the on-demand content. Set top boxes with full HD support start at $9.95/month.
While I initially was wary of the need for an STB for each TV, I found that based on the way Vz priced their service offering, my costs with Vz STB's was nearly the same as my old cable company without the STB's (I have four TV's) -- and subsequently my old cable company has gone to an encrypted QAM model as well meaning they now require STB's on all TV's as well..
It isn't clear to me exactly what you are asking.
If you cable system is analog, the TV's will all need Set Top Boxes. FiOS is an all digital service.
1). The simple answer is you need a Set Top Box (STB) or DVR for each TV if you want HD.
They are all split off the main line coax using splitters and you can have up to 7 devices connected I believe.
However I don't believe Verizon is offer SD set top boxes any longer, so I don't think you can get SD set top boxes any longer.
Your existing Cabling will probably work if Verizon upgrades the existing splitters. The STB's communicate with Verizon over the Internet, and MoCA places these signals well above those normally used for Television, so the standard splitters used by Cable Companies will not propagate the MoCA signal. No MoCA signal, and you don't get the box authorized, or program guides, or video on demand. Because the STB's utilize the Internet, Verizon will install a router even if you don't buy Internet Services from Verizon. The Router counts as one of the 7 devices.
2). If all you want is SD, you still can use a DCT-700, which is less expensive, but there is no video on demand, or channel guide available.
3). IF you have a TV with a QAM tuner, you can get a handful of channels (generally local stations) without a STB. In general all National Channels, and all Premium channels are encrypted, so you need a device that can use a Cable Card, which Verizon will rent to you, if you want to receive all of the available channels on your 'package'. I am unaware of any Televisions that support Cable Card.
Hope that helps