How long does it take to get advance parole? anyone applied for expedite processing? How long did it take? What proof did you show?

You can schedule and appointment at your nearest USCIS office and take all the supporting documents while you appear to file for an AP. If the officer is convinced about your case and the proofs you present, you'll be issued an AP the very day.
I was out to apply for one but I didn't have enough supporting documents. But the officer suggested me to bring all documents with me next time I am present and that I will be granted AP the very day.

Same Day AP has less validation period than issued one is what i heard from members of Sajha. Same day might have 1-2 dmonth validity while one done through uscis normal process would be for the duration of TPS. Is that true?

Same Day AP has mostly one entry only while the AP with the normal process has multiple entries but it also depends on the officer.