You need to do a little more archeological digging still. You need to know the whole construction of the floor before you can decide how best to install something new. I'd take up that plywood piece in the kitchen and see what's under it. It's higher than the surrounding floor, so needs to go. You should also be able to see from there what subfloor is under the original kitchen hardwood. It's too bad the space under your entryway is finished, but I'd still do some exploring from below to see what's under there, and to figure out the fix to stop it from squeaking. Is it drywall below or a drop ceiling? I know it probably seems like a pain to tear out drywall, but if you cut out a clean patch, like 2' x the width of the joists, it'll be relatively simple to repair. Just make sure to cut it down the center of the joists so that there's room to screw the patch piece to it later. Cut the hole under the area that squeaks the most, if you can. Have someone walk on and watch the floor from below to see what's moving. It might be as simple as pounding in a couple shims from below, or face-nailing a loose board from above. You want to find out what's underneath the hardwood. It might be laid directly on the floor joists, or there might be 10" wide boards that the hardwood is nailed to. Another thought is to take out a board in the hallway, since that's not original and you want to replace it. See what the subfloor is there. I am a bit puzzled at why, when you've stripped the kitchen down to the original hardwood, it's still higher than the hallway floor. Is your current surface perhaps not the original kitchen hardwood? Peeling back the plywood area will help determine that. BTW, I haven't heard of that stop squeaking product you linked to, so can't offer any personal opinion. And don't pour self-leveling compound over top of hardwood. It needs to go over plywood.

My concern is the layers of vinyl underneath the hardwood. Hardwood doesn't like sitting on vinyl and vinyl doesn't like sitting on hardwood. The two do NOT like each other. If you think this is a good idea (which it is considered on the technical side of things a horrible approach to this) then you might have to add another layer of subfloor. A nice, fat, thick 1/2" plywood to cover those two layers of vinyl. Now you can go ahead and lay the hardwood. Technically, my suggestion is hideous. It is atrocious. It is so "unprofessional" that I could weep. But if you insist on leaving the vinyl in place, then you need to think of a way to separate the vinyl from the hardwood sitting on top....and one very sure way is with plywood. Again, my suggestion is a travesty. Then again so is laying hardwood over two layers of vinyl. In the flooring world, we 'allow' ONE layer of resistant flooring (like vinyl) to sit underneath another floor. For example: the first layer of vinyl was considered "OK" to lay another layer of vinyl over top. All good. No one in the technical department batted an eye. If a THIRD vinyl floor was proposed (or a laminate, or linoleum, or cork, or hardwood, or tile, or even carpet) it would be a "NOOOOOOOO! STOOOOOP! DON'T DO IT!" type of answer. So I will officially say: No. Stop. Don't do it. There. It has been said. That being said, if you as the homeowner choose to forgo the installation instructions and the "Best Practices" as set out by the National Wood Flooring Association (as homeowners your word is "law" when it comes to things like this), you are welcome to go ahead and lay hardwood over two layers of vinyl. If you do it, I suggest a new layer of plywood to help things along - because you need FLAT. And most "old hardwoods with two layers of vinyl sitting on them" are anything but flat. Good luck. You would be better off remediating the asbestos/vinyl and getting down to bare hardwood. Now a new floor (with the help of a layer of plywood) can go down. Or you remove the original hardwood with the vinyl (this is easier to do than scrape off the vinyl...and easier = cheaper) and start with a new layer of subfloor and go after it like it should be done. But as homeowner your word is "law". If you tell a professional to "just do it", they will say "yes ma'am/sir" and do it. But they will not warranty their work. You will be left holding the bag - and all the pit vipers that lay inside. It's your call.

Go for it. I would throw down some 3mm or 6mm cork underlay down (it floats just like the hardwood) and then float the hardwood. Use a high-quality joint glue on the hardwood and you should be fine. Besides...if it gives up the ghost in 10 years, would you be upset?

Traditionally (ie. by a pro who is paid to do things well/properly) the damaged substrate is removed and replaced. Patching is possible if there are minor spots that just need 1-3mm of fill. If the whole thing has a shredded surface then it is probably easier/faster to remove and replace with the same thickness. As G&S suggests: photos please.