How do I know my bit rate, i.e. how many bits each 'real data bit' I'm having.
I'm not sure I understand what you're saying, but this is one level "above" BPSK demodulation. What BPSK demodulation gives you are bits. full stop – how you interpret these bits, and whether they have channel coding can't be said without very much additional knowledge about the data.
How can I figure out the size of my real data.
don't know exactly the corner frequency $f_c$, is it OK for me to estimate it for the carrier wave?
Don't know what that means. You'll need some way to estimate the symbol clock. For BPSK, the typical easy-to-teach approach is squaring timing recovery, where you just "collapse" the modulation by squaring the signal, and then look for autocorrelation peaks.