Replacing old wires stuck on pipes inside concrete walls with new Cat6 cable

The only way to tell for sure it to shut off the transformer and see who lost power. I obviously can't recommend that "solution".

Another problem is that even if you can tell you're on same transformer, it still won't be a good indication if it will work EOP.

The only reliable way is to plug in the adapter and try. You probably can do it for (almost) free: in most jurisdictions law allows consumers to return undamaged goods purchased remotely (eg online) for no reason. Check out if you're eligible. If it won't work, all you risk is 2-way shipping fee. Just make sure you're making the purchase as a consumer and not as a business entity.

Should be trivial; conduit (which is probably what your "pipe" is) is ideal for this.

Either use the phone wire to pull in a rope and use that to pull in the Cat6, (and use one for phone if you are still using a landline) or vacuum in a string if there's conduit, but no phone wire. If you need multiple wires, figure out how long they are, cut that (or a bit longer) and pull them all in at the same time - you'll need someone keeping them from tangling on the way in.

For single wires, if you buy the usual box of wire you can feed straight from it with no tangling.

Hollow braided rope is nice for a pull string, as you can expand the end by pushing on it and slip the wire(s) inside the braid, then smooth it down & tape the end in place. Allow about a foot of overlap so the rope can get a good grip on the wire.

Don't pull too hard or bend wires sharply (it ruins Cat-whatever wire) and where possible, pull down, not up (let gravity help you - carry the box of wire to the top end and pull down through the house.)