they both are live compatible the difference is memory storage the arcade version has 256 mb which is practically none compared to 120gb for the pro i would recomend the pro because if you want to get more memory later it will cost more than the price difference between the two also even though you can play with the arcade online the experience will be severely limited
it has nothing to do with your computer plug the cable into your router or modem
yes its true except for the live part all xbox 360 come with xbox live silver which lets you do everything but play games online live silver is free and yes you can play xbox 360 games with no hard drive but the 256 mb memory card fills up with your savrd games fast so id recomend getting one with a hard drive cuz its more expensive to buy a hard drive seperately