I recommend you have all Ethernet cables to all come to a single point in your home. It could be your basement or your office. It just depends on where you want your network hub to live. I recommend the basement, as it's easier to hide all the wires and gear. Some people like to have it more accessible, and put it in the living space; your call.
The gear involved is something that terminates the Ethernet cables and the switches. For the termination, you've several choices, including a wide variety of patch panels or even a 110 block. I installed a 110 block in my home 20 years ago, as it gave me flexibility to run phone or data over each cable. In fact, some of my old 100Mbps runs used to carry both. But those have been long since upgraded to 1Gbps only. The 110 block made it very easy to change. These days, I'd probably install an Ethernet patch panel instead of a 110 block, as I've no need for wired phone jacks.
As for the router location, Edg1 gives excellent advice. My router sits in my office. The ONT is in the basement next to my switch, 110 block and all the patch cables. The WAN port from the ONT goes up to an Ethernet jack into my office. Two additional Ethernet runs from my office carry LAN connections from the router back to the basement, where I have two inexpensive self-managed switches. They feed gigabit Ethernet to the rest of the house, including three mesh network nodes that improve WiFi coverage.
I also have a NAS installed in the basement. It's nice and cool down there year round, so it's a great place to keep this always on equipment. And, a big UPS powers the ONT, switches and NAS. Another UPS in my office powers the router while the best located mesh node has it's own UPS. So, I get to keep internet for a while during power outages.
Another recommendation is to run more cables than you think you need now. My office has five CAT-5e's and a coax going to the jack near the router. It's given me lots of flexibility. Whenever I pull new cable, I almost always pull one more Ethernet than needed. It provides a spare in case of problems or gives room for unanticipated expansion.
As for what Verizon will do, I'll be interested to learn if they will even install fios before the home is finished. They will want power available for the ONT and router, so they can activate and test the system. You mentioned electrical is done, but if the plugs aren't powered up yet, I'd be surprised if Verizon will install.
I also doubt they will pull Ethernet through your open walls. Decide where you want everything and pull Ethernet yourself, or have your contractor do it. The Verizon tech will use Ethernet cables you've installed.
Finally, know it's easy to pull cable from an unfinished basement to a 1st floor room. I've pulled many cables in my home over the years. That's why I've five CAT-5e's and coax in my office. 🙂 If you're reasonably handy, this can be a fun job and it saves on contractor expenses. YouTube is your friend.