1. Parole status if you enter using AP.
not H1-B status. You can come back to H1-B status after you do your H1-B extension or you transfer H1-b)

2. You can apply for AP when your TPS is in pending status. Actually TPS and AP can be applied at the same time.

3. Max is till the end of the current TPS period. (depends whether you opt in for single entry or multiple entry). 1 month should be fine. But you can stay longer if you need. The risk increases if you stay longer period of time.

4. Not sure. Depends. I saw someone getting it within 2 weeks. whereas someone who has waited for more than 2 months has not yet got the AP.

btw when are you planning to go for stamping?