My asylum case was refereed to imm court.I can't wait another 5 years.Can I file motion to advance???

Sure you can but the case is unlikely to be advanced. You told us nothing in your post to suggest you could be successful in advancing your court...

Can I maintain F-1 Status During Asylum Process?

You could. Situations like that you really should discuss with an immigration lawyer in private.

Each entry out of the last five entries is counted

H4 biometric

There must have been a reason why your current employer filed I 539 for your spouse. Did you ask him why?

Owning an LLC on H1B work visa?

It could be an issue since it could be considered self employment.

Can I get H1 B stamping in Cancun after changing my employer?

You need to contact the US consulate in Mexico to figure out if they accept via applications by third country nationals.

What happens if I try to open an administratively closed case before IJ?

You will go before the judge and apply for adjustment. Before you do any of it, talk to an immigration lawyer.

Does consular officers discriminate among same-sex and heterosexual marriages while reviewing an NIV ( such as H4)

They should not since marriage under the US law is gender neutral.

I have asylum pending and waiting for interview my question is can i marry some one non American citizen nor green card holder?

Your marriage and immigration status should have nothing to do with your asylum and everything to do with your decide to marry this person.

F-1 student on STEM OPT marrying another f-1 student doing masters in USA. Considering status change for dependent visa. Doable?

Everything that is legal and proper is doable. I see nothing illegal or improper presented by your facts.