http://am.nafsa.org/sites/default/files/WorkForJointF1TPS.pdf

Good news for fellow F1 refugees

Nakale,
I don't see USCIS saying they allow working on TPS EAD for F1 students. They're saying you can get TPS while on F1 but they're also saying to keep in mind that working may violate F1 status. So you've to be cautious before using TPS EAD.

"Before someone holding both nonimmigrant status and TPS chooses to work using a TPS-related EAD, he or she should carefully consider whether that employment could violate the terms of the nonimmigrant status, potentially resulting in violation of the nonimmigrant status. F-1 students who are considering working on a TPS related EAD may want to talk with their Designated School Official (DSO) and/or an immigration attorney to discuss how employment could affect their F-1 status"

I don't think there will be any problems on working on TPS EAD while on F1 status. USCIS is trying too be on a safe side by saying that because they don't want to deal with school's DSO or any other circumstances.

Does anybody know if TPS applicant( with no OPT/no f1 status) can apply for H1B or greencard ?

No man TPS is for temporary but during that time if you marry a U.S. citizen or find a employer willing to sponsor your green card then you can otherwise it's gonna last till 2016 December