Rear derailleurs have two specs - maximum cog size and chain wrap. If you're not changing your cassette, and the current RD works with the largest sprocket on there now, it will continue to work with that same largest sprocket, regardless of whether it exceeds the stated spec. Changing the crank won't change that issue at all.

But, as someone pointed out, you might be exceeding the short cage RD's chain wrap by several teeth - and my opinion is that it won't work with that much difference. But double check that spec, and make sure you have a short cage, not a medium or long cage RD.

If you do have a short cage RD and it doesn't have enough chain wrap capacity, you can easily find a longer caged RD which won't cost you a fortune, especially if you're satisfied with Sora level stuff.

As for the crank and bottom bracket: Simply find the crank you want, and get the bottom bracket specified for that crank. Shimano (and most other) cranks actually specify the BB to use. They're not expensive so don't worry too much, just get the one spec'd for that crank.

Don't worry if the crank is a 10 speed or 9 speed. If you set up and adjust your front derailleur properly, the shifting will be perfect with either. I've tried it at least a couple of times, and it really is no problem. Currently running a 10 speed crank with 8 speed derailleurs, shifters and cassette. Previous was the same, except everything except the crank was 9 speed. I don't understand why people report unsatisfactory shifting, except maybe they don't bother to check and readjust the FD when they install the new crank.