lol yes. the operating system is what keeps connects you to the hardware.
a 32bit operating system is limited to only addressing 4GB of memory (there is a work around called PAE). BUT the problem is microsoft actually limits the memory you can use, so even with PAE you still wouldnt be able to use more then 4GB of ram (unless using a the microsoft server operating system).
so for windows yes you need 64bit to use more then 4GB of ram.
Linux is a completely different operating system. it does support PAE, and thus you would be able to use more then 4GB on a 32bit linux install. And ofcourse you could use more then 4GB naively on a 64bit version of linux.
But so you are aware, if you are using windows vista or newer, you can get 64bit version. Starting with vista (and still continuing today), they merged the 32 and 64bit license keys. so that means your 32bit windows key will work to activate a 64bit version of windows. (it must be the exact same version, so windows 7 home premium 32bit English will ONLY activate windows 7 home premium 64bit English)
So if you can get the operating system files for the same version of windows but 64 bit, you could install 64bit windows and use your same key. (i think you can download windows 7 or 8/8.1 from microsoft using a product key)
Also know your ram definitely isnt holding back your gaming. lots of people have done tests using the same pc (a pc using a high end cpu and gpu), and tested with 1,2,4,8,16,64GB of ram. difference between 4 and 8GB was 0 fps. difference between 2 and 4GB was like 2 fps (1GB was unplayable).
most all of your gaming performance comes from your gpu. (just as long as your cpu meets the requirements for the game ofc).
and yes games which are online are cross platform. this isnt console lol, they all use the exact same game servers.