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.