Would it be "of the other team" or "on the other team"? The latter sounds better.
"The" player makes little sense here. What would make her "the player," rather than "a player." In Hebrew, would one refer to any player on a team as the player? Must it be ה because of של?
It would be just like in English in this case. "The player of the other team" would be maybe if two players were discussed before.
That does, indeed, work. With these context-free Duolingo sentences, it is always a bit difficult to think of the context in which the sentence could be used.
You're right, and since the necessity of "the" in some cases is different between language, this sentence is really confusing.