This isn't like DirecTV - you don't need individual cables for every tuner. There will only be a single coax coming out of your ONT. That coax will go to a splitter. From there, you'll haev separate cable runs to all your TVs. Again, you only need one cable run to each TV. For me, I just had an old-fashioned dual-lnb circular dish. There were (and still are) two lines from the dish going into my crawlspace. From there, they had both cables going to a multi-switch, and from the multi-switch they had 4 lines coming out - 2 each to my living room and family room TVs.
When I got FiOS, the installer cut the comcast cable, fed that into the ONT (it was already going into my crawlspace). He disconnected, totally, the multi-switch. For the two cables coming from the dish, he put caps on the cables, to prevent any leaking of the signal which could affect the FiOS signal. He hooked the other end of the comcast cable up to a splitter that they brought. For hooking up the cables going to each TV, the installer actually took the easy way out. I already had 2 cables going to each TV, but instead of only running a single one to each TV (requiring that he isolate one cable going to each location) he just put in a big enough splitter to handle all 4 lines coming out of it. So the splitter had 4 lines coming out of it, 2 each for the living room and family room (he just totally reused the cable that DirecTV put there).
When he hooked up my TVs, he just only used one of the cables in each location. So, I had two cables going to the wall outlet in my living room. He left both cables going to the wall plate from my splitter. But then, he only ran one cable from the wall outlet to my DVR, and he capped the other outlet (again, to avoid signal leaks).
For you, and this is tough to determine without actually seeing your setup, but if they already have cable going to your router, then I can pretty much figure out what they're going to do. They're going to cut the cable that's coming from your ONT to your router, and just use that. So they'll cut the cable, once inside your house. They'll put a connector on each of the newly cut ends. They'll then connect the two ends with a splitter. So you'll now have ONT -> splitter and splitter -> router.
Next, is your multiswitch already inside your home? If so, then it should be easy. They'll likely just disconnect the output cables from the multi-switch (those going to your TVs) and plug those into the other ports on the splitter. At the end you'll have a single coax going from ONT -> splitter, and multiple cables going from the splitter on out - one to your router, and one or two each to each TV location.
As for the HDMI cable, if you already have one in place, since you already have HD, they'll likely just reuse it. I didn't have one, so I requested one. For the TVs that don't have HDMI cables that can use them, I'd request that they provide them.