Form I-130 has Married name but travel documents do not, can I proceed with DS260 with maiden name

I agree that you should be using the married name since there was a reason for which it appears on the I-130.

No. Only execute the I864 and provide necessary documentation.

H4 Premium Processing

I539 is not one of the forms offered for processing as of now, as premium.

Applying for my marriage-based green card: how important is to have on papers the same residence address with my US husband?

Separate addresses can go a long way in negating the bona fide nature of the marriage.

I am currently on F1 Visa OPT status with "X" employer. I am getting my H1 filed. If my H1 is approved, can I join a new "Y" ?

If the I-129 is approved as subject to CAP, you cannot work for the Y unless Y is your OPT employer as well. Only when you status changes on...

Would it be possible to adjust status from TPS to Green card through spouse?

TPS is a valid non-immigrant status. However, your ability to adjust depends on the manner you last entered the USA.

H4 approved, H1B pending , nw, Consulate has asked me to come over for the third time to discuss H1b.

Consulate is always concerned with fraud and this is their job to combat it. You need to cooperate with them if you want positive outcome.

Do I need an Immigration attorney to help me adjust my status as a Syrian Intentional student in the US ?

No. I suggest you first consult with an immigration lawyer and try to determine if your status can be adjusted. Syria is not a peasant place to...

Student visa got rejected in the year July 2003 and in Aug 2003. H1B Visa got rejected in 2006.

I would not be that pessimistic. Try to gather documentation to support your employment record and other ties to the community and apply.

Breaking a contract before starting the job?

That question should be directed to an employment or contract attorney. From immigration law standpoint, you will find yourself out of status.