"Who do you like to travel with?"
Translation:З ким ти любиш подорожувати?
"мандрувати", actually, but I do agree with you. It's the first word that comes to my mind when I see "travel".
1) мандрувати is accepted in this exercise.
2) want ≠ like, so хочеш is not accepted
In standard Ukrainian we usually don't skip pronouns. In spoken Ukrainian in some cases (depending on the sentence, the speaker, the region, personal style etc.) one might skip the pronoun.