Learn Italian: Perdo la gara e chiedo la rivincita. - I lose the race and ask for a rematch.

In Italian, the verb chiedere already means "to ask for" when it is followed directly by a noun object. You do not need to add a preposition like per. You just say chiedo la rivincita (I ask [for] the rematch).

I previously learned that "to ask" is fare una domanda. Why do we use chiedo here?

You use the phrase fare una domanda when you are literally asking a question to get information from someone. You use the verb chiedere when you are requesting an item, a favor, or an action, like asking for a rematch here.

How does the word rivincita literally translate to "rematch"?

It is made up of the prefix ri- (meaning "again") and the noun vincita (meaning "win" or "victory"). So a rivincita is literally a chance to "win again", which perfectly captures the idea of a rematch.

Could I use corsa instead of gara for "race"?

You can if it is a running race, as corsa refers specifically to running or a physical run. However, gara is a broader and very common word that means a competition, match, or race of any kind, such as a swimming, skiing, or car race.