Setting up home network. USG/SWITCH/AP/MOCA Adapter?

Assuming your house coax is actually connected all over, your first pic should work.

There's a chance your Coax install may not be stapled inside the walls. If you can, get someone to check it out, if thats the case you may be able to run Ethernet along side the Coax and do a fully wired run without having to drill or cut in to drywall. I lucked out and was able to do that in my place when COX replaced all my coax lines.

If not then MoCa and Ethernet over Powerline adapters would be your best bet. The EoP adapters may be a bit simpler than a MoCa setup.

However....
From the FiOS ONT to the FiOS router...what cable is it?
Coax or ethernet?

OK, just checking.

The router gives out individual IP addresses to devices that need them.

In my case I was lucky. I live in a single story place with 3 rooms, so all the wiring is accessible in my crawl space/attic. While having my ISP check my coax lines for replacement I noticed that when the tech was hooking up something in my crawl space/attic that the cable in the wall plate was wiggling. A few minutes of pulling on the cable and then the cables in the bedrooms we discovered that the lines in the walls weren't stapled on like we thought they were. Since we were already going to replace the coax, I also ran riser rated Cat6 (taped to the coax line) along side the new coax and fished it through the walls that way. It took longer to terminate the lines at keystone jacks than it did to run the actual cable.

So basically find someone small and light enough to get up in to the spaces that run the cables and see if they can move the cables within the walls. You'll want to have someone at the wall plate/hole where the coax comes in to the room so you can see if there's any play in the cabling. If it can move, and provided they didn't drill the initial holes exactly as wide as the coax cable, you may be able to fish the Ethernet cabling along with the coax.

Keep in mind this is all going to be up to how your place was built and wired. Since you're in a two story house, you'll have firestops installed in the walls for sure so even if the coax is loose in the walls, you may have a hard time squeezing it through the internal stops. I ran in to that in the bedroom and it took some playing with the cable to keep it from catching on the lip of the hole. Since I was running coax at the time, I didn't need fishing gear and just connected one end of the old coax to the new coax and taped cat6 along with it and pulled it through the walls. Granted you could use the same trick, just hit Home Depot/Lowes/etc and buy a long length of coax (you'll need to know lengths before buying anything of course) and just run it back and forth to feed the Ethernet cabling. You'll find out pretty quick if there's enough of a gap for the cables to co exist in the walls. The downside to it all is you'll need to buy some cabling to find out if you can do it. While bulk Cat6 (make sure its not CCA grade) isn't super expensive, it's still going to be a $100-$200 or so just for the cabling depending on your lengths, plus the keystone jacks, wall plates and whatnot. So MoCa/EoP may be cheaper in the long run. It wouldn't hurt to consult the landlord before doing anything so they don't suddenly make you yank the cabling after it's all been done.