I had the same problem. Had old strip flooring that was over 100 yrs old. At one point someone replaced a section with 3/4 thick wood floor but you wouldnt know until it was tore up. When I laid all the flooring an got to the the hallway I was up about 1/2 in. I shimmed up an 1/8 in then stripped a blank of the new floor on the table saw to build a thin transition piece. There is a slight raise but everything is the same level and looks nice. So in short just use a reducer.

No idea what is good to use. We'll be moved in 2 weeks. What do/did you use to clean natural hardwood? How about travertine?

Your tile is cracked but is it loose? If it isn't actually coming off the mortar bed you could try to stiffen the joists if you have access from below. If they are traditional solid wood joists, sistering them with 2x10 metal studs to take out deflection is not that hard or expensive. You could also add a support wall mid-span. It is also possible that your floor has plenty of support on paper, but if it was put in when the house was new, the house just moved around too much because the framing dried out. The FIRST step here, is getting a tile expert on site IN PERSON to take a look at the framing under the floor and see why the tile has cracked. To get that floor up you are looking at pulling your kitchen apart, jackhammers, the works. That radiant heat is likely embedded in a 3/4" cement layer. You ready to do that? Because honestly for some framing reinforcement at $1,700 to save the floor I'd go that route unless you are ready for major remodeling.

OK...the solid hardwood is factory finished. That's fine. It means that you can choose a solid factory finish so that EVENTUALLY you can get all of this refinished at the same time AND STILL end up with the same floor throughout. What you do is find the SPECIES of the hardwood and the width (and the thickness)...and if you can find the same grade/cut. Sure you can run it perpendicular to the hallway...so long as you are NOT on wood joists. If you have wooden subfloors the direction of the joists will dictate the direction of install. Regardless. So long as you get the same species in solid, you can always sand/refinish everything at once = same floor throughout the home. You do not have to do it today...but it gives you the BEST option to do it later if you want...or just before you put the house up for sale.