Learn Polish: Ilu masz braci? - How many brothers do you have?

The word ile changes to ilu when you are counting masculine personal nouns, meaning people who are male or mixed-gender groups. Since brat (brother) refers to a male person, we must use this special form.

Doesn't the verb mieć (to have) normally require the accusative case? Why isn't the noun accusative here?

You are right that mieć takes the accusative, but the quantifier ilu overrides it. In Polish, quantifiers like 'how many' and numbers five and above force the noun into the genitive plural case. That is why we use the genitive plural braci.

What makes the genitive plural form braci so irregular?

The noun brat has highly irregular plural forms. Its nominative plural is bracia (which looks feminine singular but is actually masculine personal), and its genitive plural is braci. It completely ignores standard masculine endings, making it an exception you simply have to memorize.

If I wanted to ask 'How many sisters do you have?', would I still use ilu?

No, you would use ile. Because sisters are feminine, they do not trigger the masculine personal form. The correct question would be Ile masz sióstr?, combining the standard ile with the genitive plural of siostra.