Can I have a political asylum while holding a b1/b2 visa?

Yes you can apply for asylum if you have a valid claim of persecution in your home country upon one of the bases for asylum. As long as your I-94 is valid, even if the asylum is denied, you can remain in the US. If the asylum is denied after your I-94 expires, your case will be forwarded to immigration court for removal proceedings.

Asylum is a complex area of law.

Perhaps apply for a B-2 extension first. And, yes, a Syria national can apply for asylum, ideally before the one year deadline from date of entry to the USA.

I believe there is no reason to oppose your application for political asylum during the time of your current tourist visa, as long as you apply within a year from arrival. Also, once you submit your application, you will be able to stay here during its pendency and await the results of your petition. You will be called to the interview and the decision will be announced there, unless you receive some RFE (additional request for evidence or another sort of letter) prior to your interview - you would have to comply to that request and mail them the information they require within the allocated time. Applying for asylum is not an easy process. It requires substantial legal preparation and assembling papers.

Yes you can. Once your asylum application is filed, you can stay in the U.S. until a decision is made. Be aware that you have only 1 year from the date of you last admission into the U.S. to file your asylum application or your claim will be time barred.