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.

judiegal, your husband is wrong on two counts. First, there are square-edged prefinished hardwoods available. There is one problem with the square edge - you can get something often called "sock-catchers" if the subfloor is not perfectly flat. There are many different depths of bevel available nowadays. We had the microbevel in our previous house to avoid "sock-catchers" and it did not collect debris. I could get the floors completely clean with a microfiber dustmop. (My current house has 107yo heart-pine floors, and I am getting all too familiar with catching debris because it has gaps over 1/4" wide between many boards. Maddening.) It peeves the living daylights out of me to hear people refer to engineered hardwood flooring as not "real" wood. It is just as much "real wood" as solid wood. Call solids solids if that's what you mean. We chose an ashwood engineered flooring for our previous house because NH has massive swings in humidity (we did not have central AC - it would have cost well over $12,000 to install) which can create problems with cupping, buckling, or gapping, and were extremely pleased with it. I cannot TELL you how many dealers and installers tried to bully us into site-finished flooring, even though we could not afford to vacate our house for a week plus, and there was no way we could stay and try to work around the construction and finishing (I would have almost surely been very sick from the stain and finish odors, and let's not even talk about trying to keep the cat and dog hair out of the finish.) Here in New England, the attitude of "we've done it this way for 100 years, we're not going to change now" is rampant! We did not have any problems with the floor delaminating from cat barf, spilled beverages, tracked-in snow, that sort of thing, and the aluminum oxide finish was very durable. The manufacturer, Robbins, did NOT recommend putting any additional finish on after installation. They were nailed down, not glued (the installers checked with Robbins to make sure they could be nailed instead of stapled). The reason why johnatemp thinks she can always spot engineered flooring is because CHEAP engineered floors use a rotary-cut veneer, which looks like plywood; better-quality engineereds have sliced and even quartsawn/riftsawn veneer layers. Another frequent myth about solid wood flooring - it cannot be refinished as many times as people are led to believe. You can only sand them down to the level of the tongue, which is usually less than a third of the thickness of the board. Old (50+ years) floorboards can often be flipped and reused if very carefully pulled up, but that is quite rare in modern flooring. Our floors were warranted for three full refinishings; the veneer layer was a third of the thickness of the board. However, you don't need to completely refinish a hardwood floor unless you have deep gouges or want to change the color. These days, if you want to refresh the finish and remove small scratches and scuffs, you just do something called a screen and recoat. It costs a LOT less than a full refinishing, too. Here are a couple of pictures. The floors are, as I said, from Robbins; style is and color is Tigra.

Choosing the right flooring type isn't enough for your living condition. You will have to consider the types of cut: flat sawn, rift and quarter sawn, quarter sawn or live sawn. Quarter sawn and rift & quarter sawn will be best choices, because expansion and contraction is along the thickness of the wood. You will need climate control all year round. 70 degrees at 40 % relative humidity is where you want to be throughout the year. The use of a humidifier/dehumidifier will be needed year round. Although, engineered flooring is more stable than solid. Using the wrong wood specie in your living condition can run the risk of de-laminating with engineered wood. If, the top veneer is more or less stable than it's core and backing. You will experience de-laminating in extreme humidity swings. Take in account of what the engineered flooring is constructed of: mdf or plywood. The width of the flooring will account for how much expansion and contracting also. 2 1/4" strips are the most stable. As you go wider, expect to see more wider seasonal gaps. Proper acclimating is critical, should be performed with a moisture meter. For 2 1/4" strip flooring , sub-floor and flooring moisture should be within 4%. For all other widths 2% moisture differentiation is allowed or should not be exceeded. Don't forget proper expansion gaps, which is the thickness of your flooring, example: 3/4" thick = 3/4" gap. Sub-floor type and condition will affect the performance of wood floors. All this needs to be considered for the proper and successful performance of your wood floors.

I don't think the hardwood from today is nearly as good as the hardwood from even a few decades ago. I have small dogs - both less than 20 pounds. But they're young, like to play, and they've left some pretty amazing scratches and skitter marks in the hardwood - in 2 different homes. Yes I keep their nails trimmed, they get a quick dremel every week. The old floors were site finished oak hardwoods. And there were spots where the scratches were getting pretty bad. The one house in question is in a beach town, so between the dogs and the sand, the floors were very scratched up. I deliberately chose a wood look LVP and it's held up incredibly well, but no one will mistake it for site finished hardwood either. I have engineered wood in my primary home - and while they've scratched that up too, it's somewhat less noticeable because it's a matte finish, and it's pre-finished engineered, which I've heard may be somewhat harder.