After I get my EAD, can I travel outside of the US and return to the US?

The EAD will remain valid, notwithstanding travel.

My question is: Can I declare my parents as dependent in W4, even they stay in other country?

This is probably not permissible under US tax law, but you should consult with a CPA.

Why is it that it is rare that an immigrant sues their sponsor for support? Is there some deterrent keeping them from doing so?

It can happen, since the affidavit of support is contract between the sponsor and US Government for the benefit of the sponsored immigrant.

If you were lawfully admitted to the US at some point, it should be possible to refile for adjustment of status, if otherwise eligible.

Is it okay to travel outside the U.S. and get the H1-B stamp in my passport renewed while my criminal case is in progress?

You could be stuck outside the US for a long period.

Starting a new Business

Is that a question?

Can I assist my fiance in obtaining documentation if I assisted my ex-husband 3 yrs ago?

You can expect the validity of both marriages to be questioned with heightened scrutiny.

Can my father become US citizen even when he spent more than 9 months out of US back in 2009?

Yes, he can. he should consult with an immigration attorney to discuss his case and ascertain the earliest date on which he can file.

In what context?

Citizenship threw asylum after visting home for 2 months.

Such travel can raise questions as to the validity of your underlying asylum case. Consult with an immigration attorney for specific advice.