>Isn't it more significant not where they emerge (which is by random mutation) but where they do well?

Random mutation happens more when we give it more chances, by letting it infect more unprotected people. We'd rather avoid that, so it's also important.

>Is it even particularly easy to say where a variant originated?

No, but we have gene sampling to help.

>Could the mutations happen multiple times independently?

They can and sometimes do happen independently.