Learn Farsi: بعد از اینکه بامبو را کاشتند، برکه را تمیز کردند. - After they planted the bamboo, they cleaned the pond.

You need اینکه because a full clause with a conjugated verb (کاشتند) follows. If you were just connecting to a noun phrase, like 'after the planting', you would only use بعد از. The word اینکه acts as a bridge turning the clause into something the preposition can attach to.

Both verbs here are in the simple past. Could we use a different tense for the first verb since it happened before the second one?

Yes! While using the simple past (کاشتند) is perfectly natural for narrating consecutive events, Farsi also frequently uses the past subjunctive or the past perfect after بعد از اینکه to emphasize that the first action was fully completed before the second one started.

Why do both بامبو and برکه need the specific object marker را?

Using را indicates that these are specific, known items in the context of the story: the specific bamboo they were working with, and the specific pond they were tasked to clean. If you dropped را, it would sound like they just planted some random bamboo in a general sense.