A new Wifi router would not be the best solution. If there’s an available conduit between the houses I would go for fiber install and get 2 ethernet to fiber adapters then get plug that into your current router or look at directional antenna assuming you have line of sight between the houses.
Not without better cooling systems. Intel and AMD CPUs for higher performance can crank out performance until they hit the thermal limits.
You can do things such as Eluktronics where they have water cooling to dump the heat out faster then heat can be generated.
Batteries are going to be an inverse relationship with performance. The better the CPU the worst the battery life is going to be.
If you are running just generic stuff then a Mac will meet your needs. If you need to run games / Windows only applications then Windows laptops and desktops will meet your needs.
All rooms having dual cables pulled to each room with CAT7 wiring.