furke is right !!!

I will try to write what he\she has alreday posted out there...

If we get married now, what will be the case?...
There is no case right now. Ur GF doesn't have greencard yet. Even to travel she has to get 'travel documents'. Since your are in F-1 you still have to maintain ur status.
This is true even when she gets her GC and ur married to her and she files ur GC (I believe USCIS form I-130). Again, u still have to maintain ur status while she only has greencard. Things would have been differnt if she was a citizen.

Can we file for GC at the sametime or not?
As furke said - "NO"; you have to be married at the time of filing asylum.

Also, I was wondering where did you get the info?
My information - google, USCIS websites, radio, forum, friend's case, lawyer response... etc...

is there a genuine site for it?
Check USCIS or GOOGLE forum; there are hundreds out there....