Learn Hardcore Russian: Я не отключу интернет до тех пор, пока не отправлю отчёт начальнице. - I won’t turn off the internet until I send the report to my boss.

Russian regularly uses double negation in sentences like this. The main clause is negative (не отключу = I won’t turn off), and the subordinate clause after пока is also typically negative (пока не отправлю = until I send).
It doesn’t “cancel out” like in some logical interpretations of English; it’s simply the normal Russian way to express “not until”.

What’s the difference between отключу and отключаю here?

Отключу is perfective future (a single completed action in the future): I will turn it off (once).
Отключаю would be imperfective, usually meaning either a repeated/habitual action (I turn it off) or a process (I’m turning it off), which doesn’t fit this “one-time future after a condition” idea as well.

Why is отправлю used (perfective), not отправляю / буду отправлять?

Because the meaning is “until I finish sending / successfully send the report.”
Отправлю (perfective future) focuses on the completion of the sending.
If you used imperfective (буду отправлять), it would lean toward “until I’m in the process of sending,” which is usually not what you mean in this type of deadline/condition structure.