I had the same problem. Had old strip flooring that was over 100 yrs old. At one point someone replaced a section with 3/4 thick wood floor but you wouldnt know until it was tore up. When I laid all the flooring an got to the the hallway I was up about 1/2 in. I shimmed up an 1/8 in then stripped a blank of the new floor on the table saw to build a thin transition piece. There is a slight raise but everything is the same level and looks nice. So in short just use a reducer.

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.

Thanks for the encouragement. Repainting was vetoed by DH after noting the wear-and-tear of our offspring and also the lackluster performance of our chosen paint, dry time, cure time on a topcoat, etc. We loved having a colorful floor, though, so after looking at (maybe everything?) we, too, have decided on roll-out flooring. It's a fiberfloor - DuraFlex, which is exclusive to our retailer here in VA, but exactly like the Tarkett and Armstrong & etc. We didn't want to fight the rest of the wood fllooring throughout the house, so we chose a stylized tile print that hopefully will look very very "on purpose." I wonder what kind of a transition is good for the lowest profile? Metal, perhaps? I will post again after receiving and installing our fiberfloor. Here's hoping the roll-out treatment is the most forgiving for the floors!

My dad loves parquet. I am not a huge fan but I don't hate it. Seems like it often elicits a love or hate response in some folks. I would be worried about that only if resale was going to be an issue in the future not too far off. Otherwise, you have lots of options, maybe even hardwood laid over the tiles? I'd check to see if that was possible. There is something to be said for a small house with uniform hardwood floors throughout, I think it is a great way to go if the house is small. Other options are cork (my brother did this) or linoleum, although I think cork would be better on the feet. It seemed like kind of a pain to install, but he likes it.