License

If you can legally obtain a driver's license from any state, you can use it to drive in other states.

Yes, you may first apply for TPS and put off applying for TPS EAD until later.

Adjustment of status or Consular Processing for a spouse? what's the best option in my case?

You did nothing illegal by marrying 2 weeks after entering on B1/B2 visa. You will, however, have a problem if you try to adjust as you may be...

I-864 Affidavit of support

Who was your I-130 petitioner? That person is your sponsor and must submit the affidavit of support, I-864,

As long as you have work authorization, you are fine.

Can the fact that I extended visa cause me any problem at the border, while traveling to the USA again?

if you did not overstay or otherwise violate your status, you should be ok to reenter if you have a valid visa.

I am a student undertaking a master course in Boston. I have won the DV lottery and I am on j 1 visa with a 2 years home reside

No, you must first either complete or obtain waver for the 2 year home residence requirement.

H4 CONVERT TO H1B

Your Je EAD will have no effect on your H1B application.

Can I leave the US for a short time while my EAD RFE is pending?

You may travel with a valid j1 stamp.

Is it possible to make two cash in same time Asylum and sponsorship?

yes , but you must be in status to qualify forCOS.