Can "סוף סוף" be placed after "אנחנו"? If not, how would you say "We finally win."?
It can be.
Is there a reason that "we are finally winning" would not be correct here?
No, it should be accepted.
I associate סוף סוף more with "in the end" or "at the end of the day" than with "finally". Am I mistaken about this?
I do think סוף סוף means finally, if you want to say "in the end" that would be בסוף or בסופו של דבר.
A Hebrew translator I use defined finally as סוף כל סוף". What's the difference between that and "סוף סוף"?
ll סוף כל סוף is also correct; it's more formal and more emphatic.