Peel and stick tile directly over hardwood

You're right that an underlayment would require nails which would probably ruin the floor. In principal, you can usually get most of a peel and stick tile up with a heat gun and a spatula. After that the floor would need more sanding to refinish than otherwise to get up any adhesive or stuck hard tiles.

Remove the tiles with something harsh like a 6 foot scraper bar and the gouges you leave might ruin the floor at that time. Vinyl tiles will ruin the surface finish but the wood underneith should be fine as long as removal isn't too rough.

The obvious other choice is either area or wall to wall carpets. That tends to preserve the floor as long as the installer dosen't go crazy stapling the pad down. Small holes in a wood floor usually fill in somewhat when refinished. I hardly notice the old staple holes or the tack strip damage along the edge, mostly covered by baseboards.

The previous owner of a house we bought had done that exact thing. It held up fairly well over the few years we left it in place (the boards were nice and tight). Sanding for a refinish was not a problem when we removed the tile.

IMO, you will do a lot less damage to the HW this way and later you or the new owner can enjoy them again.

Wet mop the floor with an ammonia cleaner (like Parsons), rinse and allow it get very dry before you install your tile.

You can refinish it if you cover areas dogs go with butcher paper, heavy paper on a roll for a few days- to a week, but it depends on their nails and habits, it worked for me, Ive seen dogs trash cheap peal and stick in a year. It depends on the dogs.