This question is way too complex for an Internet site. Your friends needs to discuss in person with a lawyer.

Filing for my unconditional green card through marriage. I will be filing for my unconditional green card next month.

YEs, this is a huge problem.

TPS, Advance parole, AOS

I answered this one already.

Following I-131 route, I will loose all of my time / residency already gained. Is there any success story and how?

Go see someone in your area. You will not get anywhere on here.

I just got married to my wife she applied for welfare because of hardship will it be a problem to get my green card .

Well, your wife does not make enough money to be the financial sponsor so unless you obtain a joint sponsor it will be a problem.

You may be able to file for TPS if you carry a Syrian passport. Asylum can be filed and you will need to argue change of country conditions.

My wife has to travel back to home country after filing EAD. Will there by any problems?

You will need to provide more details before anyone can answer this question. EAD has nothing to do with a travel document.

How can i become a legal resident?

If you leave, you will not be allowed to return for a very long time.

Czech and US dual citizenship

This question is best answered by a Czech lawyer but it does not sound like you can do it based on the information that you provided.

Is it possible to sponsor my boyfriends visa?

Unless you are an employer or want to marry, there is no way for you to sponsor him.