Do not panic! If the doors are painted, it's EASY ... 1 - remove door 2 - sand or plane off the paint on the bottom of the door, and strip and sand at least the bottom couple of inches of the door. 3 - glue and nail (with finishing nails) a strip of wood to the bottom that is as deep as you need. Cut this to fit from any nice clear-grained boards. Depending on the height you need, it can be plain 1-in boards up to 2x6 Fill the end grain of the board, let dry, sand smooth. 4 - sand the joint and the rest of the door lightly, prime and repaint. Mucho work, poco money required.

Of course the easiest way to do what you want to do is to install a different width in the kitchen and use something to act as a transition between the two floors. If the kitchen section will be at the same finished height and you can and want to weave new into the old, then you will have to have oak milled to your 2 1/2" size. Not impossible, but will cost more than factory runs at the usual widths. I don't have pics for you, so I hope someone does and can post them for you.

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.

What type and size hardwood? Solid or engineered? Prefinished or site finished? Where are you located (labor is very region dependent)? What amount of square footage? What floor of the house? What of of subfloor and what sort of prep will be required? New construction or existing house?