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 mother is 85 years old and has Parkinson's, so she has trouble getting around. I am keeping her in mind with your question. I would go with a low pile carpet, if you were my parent. It is softer for when she falls down. Mom has hardwood in her foyer, dining, and halls, and although they are beautiful, they are mostly covered with throw rugs, which are a tripping hazard, but less slippery for her. She has trouble with getting a "grip" on the floor when standing up, and carpet helps with that. Wheelchairs will run grooves in a hardwood floor.

That hardwood looks to be red oak with either oil based finish that has oranged (ambered) over time or a light stain. The SPECIES of the hardwood is pretty easy to match. If you are going to add more hardwood, then the SPECIES match is the most important thing. You can always work with a different COLOUR on the upper floors. That is completely acceptable. And as Sina points out, you can always do the hallways/master bedroom in the hardwood and then run carpet (durable as you can find) in the other areas. Again this is perfectly acceptable. Many, many people prefer carpet in bedrooms and offices. Because the hardwood looks to be red oak in a 2" - 3" wide plank, this should be super easy and (relatively) cost effective to find/install. You can save money by adding carpets to the other areas. Red Oak can then be added to other rooms at a later date if you feel you want a house that is wall to wall hardwood. It is one of the easier species to find/install/match. They key to this is the species match.

Definitely choose wood. Vinyls are great, however, for resale value, wood is the absolute best ROI especially for the main living areas. You can choose an engineered or solid- whichever makes most sense. We are able to provide engineered HW that can be sanded and refinished just as many times as solid. If the wear and tear is something you are worried about, consider choose a floor that has a reclaimed look with a strong grain pattern and saw markings that can hide many of the scratches and discrepancies that will happen over time. If that specific design isn't your style, consider a light color floor. Oiled floors are a decent choice because fixing superficial scratches are relatively easy. Oaks are good because of their graining and if kept a light color- will be able to disguise many of the scratches. You can ALWAYS sand and refinish your floors a few years down the line if needed. Plus, when you go to sell and the floors have been lived on- they can be sanded and finished for the new buyer. A vinyl floor will need to be replaced. Hardwood is more expensive at the moment, but they actually end up paying for themselves years down the line with the proper care and maintenance. Or you can choose wood in the dining and great room and a tile in the kitchen. SJ McCarthy has a good recommendation for hickory and oak. Another option is Character Maple(make sure it is hard maple). If you really want to be bold- go for Brazilian Pecan. :)