I have a GA H55M-D2H motherboard(i think it is 32 bit).Recently i came to know that my processor(Core i5-650) is a 64 bit processor...so can i install a 64 it operating system and RAM more than 4GB?
Your board has no impact on if you can use 32 bit or 64 bit, that is all determined by the CPU these days, and since all the CPUs that board supports are 64 bit it must support a 64 bit OS.
I dont think that board is limited to 32 bit, it supports up 8GB dual channel ram so you wont have any issues installing a 64 bit OS. Check the link for RAM frequency support, there is a note if you want 2200+MHz speed http://www.gigabyte.com/products/product-page.aspx?pid=3572#sp
Your board has no impact on if you can use 32 bit or 64 bit, that is all determined by the CPU these days, and since all the CPUs that board supports are 64 bit it must support a 64 bit OS.