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.

I learned my lesson and wish I went with the thicker engineered wood like I did in 2009 as it is more sturdy, feels better to walk on and is warmer. I had to redo my engineered wood floors due to damage from my water heater.

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.

For 3/4" x 5" , You will need to do a nail assist installation, which is a full troweled glue as your main fastener and cleats or staples, spaced 18-24" as secondary fasteners. This is standard installation practice for wide planks. The alternative is to do a glue assist, which is the opposite of a nail assist and much more tedious. Beads of glue is applied in a continuous "s" pattern on the back of "each" flooring board and nails/staples are at 6"-8". The nail assist is a better and faster install, but not the cheapest. With the correct type of adhesive, the adhesive acts as a vapor "retarder" , not trapping moisture on or above grade installation. More prep work is involved. If, the installer does not have familiarity with adhesive types. Have them use the glue assist method to stay on the safe side. Or find someone who is.