"Last Wednesday I went running with my girlfriend."
I think you’re right that you could. Hopefully you reported it so they can increase the set of correct answers:
The past tense is indicated by the fact that it happened in the past; no need for 了。
It's not correct 去 mean go as in go to some place, not go as in go do something.
Is it possible to swap 我 and 女朋友 in this sentence? Would be correct to say 女朋友和我 instead of 我和女朋友?