I was under impression that Dame meant lady and Frau meant woman. What is correct translation?
More or less, but German words are not obliged to be applicable in all the same situations as their nearest English translations and vice versa.
Apparently they are on Duolingo, given the behavior of some of the sentences that contain Dame...
travelling was given as one of the translations of wandern and yet was rejected!
Does 'wandernsehen' work as a 'separable prefix' verb here? I'm not sure I understand why 'wandern' is in infinitive.
"wandernsehen" is no existing verb. You simply have two verbs here, wandern and sehen, just like see and hike in the English sentence. Now in German, you put the first verb in the flected form according to the subject ("ich sehe") into second position, and the second verb always comes at the very end of the sentence in infinitve. It's similar with seperable verbs, where you just put the seperated verb part at the end.
In German, the verbs are pretty easy to place, even if you have four verbs in a sentence, you will always have the flected one in second position and all the rest at the end in infinitive:
Ich hätte ihn mit dieser Dame wandern sehen sollen= I should have seen him hiking with this woman.