How to proof, B2 visa, who unlawful overstay in US but for less than 180 days can change their status through DV Lottery?

Depending on what stage your asylum application is in, you may or may not have accrued unlawful presence therefore needing a waiver of...

DS160..."have anyone filled immigration petition on your behalf"?..what should I answer?

The answer should be no since the I-130 was never filed on your behalf.

I'm on F1 visa-2015(Opt status). Can I apply for a H1 visa & green card. How it affects my admissibility to USA?

You should consult an experienced immigration attorney to review your criminal records and provide proper guidance before you travel.

I797C extension on green card: what date does extension begin?

One year from the date I-751 is received by USCIS.

Admit until date on the entry stamp and Form I-94 does not agree with the ending date on the I-797 Approval Notice (H1-B)

Does your passport expire before Dec 30, 2017? If the answer you can point that to the CBP Officer.

Category (c9) if you are filing with I-485 adjustment of status.

Automatic OPT Extension after a H-1B Cap Exempt Petition

I would advise premium processing of the H-1B so that you receive an approval before your OPT expiration.

Can I work [on F1 OPT (expiring June 14th) until i get my H1b rejection notice in Mail? supporting Docs ??

CAP gap is extended until you receive a rejection notice. You may check with your DSO for confirmation.

You have 180 days to file the I-140 after PERM approval. If your priority date is current, you may concurrently file the I-485.

Parallel green card filings

We do not have all information here for example what category will either I-140 be in? EB-2 or EB-3 and who will have the priority date current?...