The way to future proof is a conduit, so that you can pull through whatever gets invented in 20 or years time and your cat 6 is obsolete.
Trying to second guess that cat 6A might be able to support some 25GB future standard is a fools errand. In the OPs situation, install cat 6 in conduit and if you need to pull through something else in 20 years it’ll be an easy job.
Mains voltage, lease of all your American (I assume) 120V does not cause issues for Ethernet in any practical domestic setting. Even with European 230V it’s not an issue.
The whole thing about power causing interference is only a data centre issue.
Just route your cables as makes the most sense. If that means crossing or using the same conduit, that’s fine.
Any old router that you have lying around the house, or can be picked up locally secondhand for no or minimal cost can probably be put in access point mode.
I don’t see the question as "what do I need’, but “how do I make it work with what I’ve got”.
The access point I have in my garden office, for example, is an old TPLink AX50, which has done it’s time as a router and is now an access point. Nothing wrong with it.
I know power line is bad in many ways. It is also the most simple, so depending on the use the OP has in mind it might be cheap, easy and suitable.