بعد از آنکه

The only difference is register. 'ān-ke' is formal/written, while 'in-ke' is informal/spoken. They mean the exact same thing.

Yes! It is very common to start a sentence with it to set the time frame. Just remember to put a comma after the first clause.

Usually, the Simple Past (raftam) or Past Narrative (rafte-am) is used because the action is completed. However, for future plans, the Simple Present can be used.

Neither is 'better', but 'پس از آنکه' is considered 'purer' Persian because it doesn't use the Arabic word 'بعد'. It sounds very literary.

You can, but it might look a bit too formal. 'بعدِ اینکه' or 'بعدش' is more common in SMS.

Yes, it specifically refers to temporal sequence. It doesn't mean 'because' or 'although'.

You would say 'بعد از آنکه بروم' (formal) or 'بعد از اینکه برم' (informal).

No, in this specific structure, 'ke' is mandatory to connect the following clause.

The opposite is 'قبل از آنکه' (ghabl az ān-ke), meaning 'before that which'.

In formal Persian, 'that' (ān) is used to refer to an event as a completed, distinct point in the past.

Yes, it's very useful for polite sequencing, like 'After you enter, I will follow' (بعد از آنکه شما تشریف بردید، من می‌آیم).

Yes, though poets often shorten it to fit the meter, the logic remains a staple of Persian poetic narrative.