Mulberry:

Cherries don't get fire blight so no concern there. It might be better planted 10ft away or at least a few feet. But for us backyard growers other considerations usually prevail. So I'd say plant it wherever is best for sun etc. Do remove all of the apple stump that you can if it's rotted. But again I won't spend $$$ removing a solid stump.

Thanks! Good to know that cherry trees aren't susceptible. I thought since they are somewhat related to apples they might suffer too.

With cherry, you need to look out for bacterial canker. It is rather common and troublesome.