My opinion, from a similar situation, use the coax to pull cat6 through and replace it. Assuming it's not stapled down or too tight. In a previous home I rented I didn't want to run cable knowing I would only be there a few years so I picked up MOCA boxes; lets you run eth over coax. I connected a switch in the bedroom for a steam box and smart TV to the router downstairs using this and worked fairly well but never did intensive bandwidth tests. Might be worth looking at.
There is also phone line to every room but I am not getting anything done on 56k lol
Depending on when the house was built, the phone line might actually be ethernet without using all the wires. Unscrew the plate and pull it out and inspect. If it's CAT5 or something, it's as simple as swapping out the plates and splicing the lines.
MOCA is also an option as the other poster said.
My other question is, what is the easiest way to adblock everything at the network level, so that no devices need to run an adblocker of their own? Should I pass the connection through a small computer, or can I run it on something else?
PiHole! I run mine on a pi zero that gets both wired ethernet and power from the router using this adapter. I spliced my own tiny little ethernet cable and used some velcro pads to attach the pi zero to the router.
Set the DNS server on your router to the IP of the PiHole and it'll filter ads based on the block lists that you can customize. The downside to DNS adblocking (which is the only network-wide kind) is that it can't block ads that come from the same source as the media you're trying to watch. YouTube is an example of this - all ads that run on YouTube are served from the same domain that hosts the videos, so it's impossible for the PiHole to know what is what.
Highly recommend keeping local ad blockers on devices because they can filter much more than just DNS.