Learn Portuguese (Portugal): Nós já não fazemos noitadas.

Adding já to não changes the meaning from "we don't" to "we no longer" or "we don't anymore". It emphasizes that this is a change in behavior: you used to have late nights, but that state has officially stopped.

Could I use ainda não instead of já não in this sentence?

That would completely change the meaning! While já não means "no longer" (an action has stopped), ainda não means "not yet" (an action hasn't started, but might). Nós ainda não fazemos noitadas would mean "We don't have late nights yet."

Why is the verb fazer (to make/do) used here instead of ter (to have)?

While English speakers say they "have a late night," Portuguese uses the expression fazer uma noitada (literally "to make a late night"). It is a set phrase or collocation that you just need to memorize.

What exactly counts as a noitada?

The word noitada comes from noite (night). It usually refers to a long night out partying or socializing with friends. However, depending on the context, it can also mean pulling an all-nighter to work or study.