Well, The reason I wuold suggest is because when you get B1/B2 visitor visa I presume that you got atleast 5 years or 10 years long visa, which allow you to stay for 6 month pervisit in USA. If you have this type of visa valid, and when you apply asylum and been rejected, the USCIS do not reject the visa you already got. So it is very logic to understand that you can still use them for re entering and traveling from USA. They did not revoke or reject your previous visa but denied your asylum case for immigrant visa, so in this case I am quite positive to say that if the visa sticker is still alive and valid it should not be a problem to enter again unless they inform you about your visa has been revoke......But I am not a lawyer please understand it is just a logical thinking...
In regard to F-1, I also think again if the visa sticker is still valid, and your I-20 is valid you should not have problem to re-enter USA. But at the sametime the Immigration Officers are the best people who can tell you the perfect answer...
But if yuor F-1 visa is expired but valid I-20 form, if you travel out you will need re-entry visa sticker stamp in your passport in the US embassy out side of USA. But will they reject the visa? Possible since you are applying fresh visa now and already made an attemp to immigrate....and your A number probably appear in their system.
After all, if you scare of being unable to come back to USA and want to live in this country, you should think about other type of visa that would help you out to legally obtain permanent, such work visa H-B1 B2 EB12345, marriage or anything that helps.
And moreover, your F-1 visa is not rejected when your asylum was rejected so you can stay as long you have F-1 visa in the USA. If your purpose is to study, you can continue to study as long as you are enroll in the university. So why warry? with F-1 you can study full time and work part time which is already very good...