What issues could one encounter when leaving the US as an illegal alien?

There is generally no problem to leave the U.S. If you are near the Southwest border, I have heard the deport people who are on their way, trying...

F1 married US citizen & seeking AOS. However, husband is unemployed

You are fine, your counsin can sign the joint affidavit of support.

What would be the basis for you to file for a green card? Just having an E2 would not make you eligible.

What kind of support can I get from my husband.

By law he is obligation to support you at 125% of the poverty rate. You should file suit to enforce the form I-864 affidavit of support against...

Can I fly domestically within the U.S. (S.F. to Dallas) with only a Mexican Passport (no stamps) and Matricula Consular I.D?

There is a considerable risk that you may be put in removal proceedings if you are not in status.

Judge denied asylum before 180 days pending

No.

Immigration weiver601 family unite whan final rule

If you are asking about stateside processing, it is expected in about January or so.

In the eyes of immigration authorities, is my brother's case considered terminated or administratively closed?

If the former INS border agents denied his application for admission, the case is terminated. He is, by the way, very likely barred for life from...

I have an offer from a US company and they are transferring my H1b from previous employer (never used and expired in 2011 Oct)

No, you cannot port to a new employer after your visa is expired, or if you are not currrently in status working for that employer.

Work authorization denied - Immigration law

Until you receive the denial, explaiining why your application was denied, there is nothing to do. Incidentally, if your application for adjustment...