Learn Russian: Извините, сколько времени?

The word сколько (how much or how many) is a quantifier. Just as you learned that the numbers 2, 3, and 4 require the genitive singular, сколько also forces the next noun into the genitive case. You are literally asking how much of time.

I thought neuter nouns in the genitive singular ended in -а or -я. Why does время change to времени?

время is one of a few special neuter nouns that end in -мя instead of the usual -о or -е. When these special words change their case, they add -ен- before the ending. That is why its genitive singular form is времени instead of just adding a regular ending.

Are there other ways to ask for the time in Russian?

Yes! Another very common way is to ask Который час? (literally, which hour). Both сколько времени and который час are perfectly natural and you can use either one.

Why does Извините end in -те here?

The -те ending makes this a formal command. You use it when being polite to a stranger, using the formal you (Вы), or when addressing multiple people. If you were asking a close friend, you would use the informal version Извини.