This sentence has to start with the verb. Pāʻani is the verb 'to play" , and pōpeku elaborates as to what kind of playing they are doing. the sentence is correct.
"Pā'ani kākou i ka pōpeku ma ke kau ho'oilo" should be accepted. That's the format used in the "sports" lessons.
You are correct, and there is nothing special about football as a sport.
Pā'ani kākou i ka pohili i ka hopena pule.
"We play baseball on the weekend."
I do not know why DL insists on the compound verb for this prompt.
Why does popeku follow pa'ani. Popeku is the objecty of we play. I don't agree with your answer.