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.

We are installed a red oak hardwood floor in our kitchen. I'd like to put a rug of some sort near the stove and sink to catch spills, water, etc. What backings won't harm the finish? I've heard that latex can stain or even get stuck. I could buy a washable full-fabric scatter rug, but then I'd have to worry about slipping. Any ideas?

Shipping is expensive, but I received them very quickly. Backings are made from latex and nitrile rubber (Multi-grip), see the link for a description of two types. They stay very flat, no curling, and have a very low profile. I have them on all three of my exterior doors. The great thing about them are that they can easily be machine washed in a normal washer and drier. They are not bulky! I went through a couple of those expensive gel mats which both curled after a couple years which made them look bad.

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.