Cat 6 everywhere, if you have a furniture diagram all the better - no sense putting in a jack on the other side of the room where your desk will be. Figure out where your furniture will go to and run at least 1 jack there, most of the cost is labor so I'd run 2 jacks to each room, just like outlets, (almost).

I can't read all that but check duplex status on relevant links.

Much better than a weak wifi signal and a solid option if a hardwired ethernet isn't practical.
pick your brand.

https://www.amazon.com/Powerline-Computer-Network-Adapters/b/ref=dp_bc_aui_C_5?ie=UTF8&node=1194444

No, don't go mesh, especially when you have a choice or a chance to change it. Mesh can work in many circumstances but it's like the #3 option are far as performance and reliability goes. An there aren't many more options than3 or 4.

You fucked up and opened your shit. Running your own server to the world is a big job needing a firewall.

Maybe, I work in enterprise WifI and we'd never use mesh, really all because of the unknown interference. A homeowner could have a wifi TV on the other side of a referigator and get a garbage signal because of the referigator. Re-word and say "if you get lucky"..

No, just don't. Mesh might work for a doorbell, but just no.

wow, that looks DIY.

It'll be fine for 10 years or whatever. Put it in a conduit/or cover it with a riser if you're worried. It's the UV/SUN damage that will be the long term cause of concern.