Are x64 registers interchangable, in the sense that any instruction that works with one combination of them will work with any other? Is there performance difference or any other considerations that make them different from each other, apart from the names?
There are some restrictions, and some differences in encoding.
rsp (and esp, etc) may not be used as an index register. There are many instructions which take arguments or return results in particular registers - for example, the variable shift instructions only take their argument in cl.
The arithmetic instructions (and test) have short encodings for rax plus a 32-bit immediate:
8: 48 05 ff ff 00 00 add $0xffff,%rax
e: 48 81 c3 ff ff 00 00 add $0xffff,%rbx
I'm sure there are some other bits and pieces I can't bring to mind at the moment: consult the architecture manual for the gory details.