Learn German: Hast du ihn beim Spiel gesehen? - Did you see him during the game?

beim is a combined word made from the preposition bei and the article dem. Because bei always requires the dative case, the neuter noun das Spiel changes to dem Spiel. In German, it is very common to squeeze bei and dem together into beim.

Doesn't bei usually mean "at" or "with"? Why is it translated as "during" here?

You are right that bei is often used for locations or people. But when you use bei with an event or an activity, like Spiel (game), it becomes a time word. It means the action happened "during" or "at the time of" that event.

Why is it ihn and not er or ihm?

ihn is the accusative (direct object) form of "he", meaning "him". We use the accusative case here because he is the person being seen. The subject actually doing the seeing is du (you).

Why is gesehen pushed all the way to the end?

This sentence uses the spoken past tense, called the Perfekt. In this structure, the helper verb (hast) takes the first position for a yes/no question, and the past participle (gesehen) is kicked to the very end of the sentence.