Can I install my old OSX on a different computer after upgrading?
Yes, depending on the hardware of the other computer and the type of Snow Leopard disc you have. For example, if your existing computer is an iMac, and the other computer is a Mac Mini, and the Snow Leopard disc you have is the disc which came with your iMac this might not work. Many OS / recover disc which come with a Mac will only work on that model. The only way to know if this will work is to test it.
When you install Mountain Lion, you will have the option to Upgrade, keeping all your applications, users, and most of your settings, or Archive and Install which will keep a copy of your current system (which you cannot easily roll back to) but do a clean install. You can also Erase and Install where you will lose everything. OS X upgrades are fairly safe and while you might have to re-install a few applications (VMWare Fusion and Adobe products come to mind) you are unlikely to lose any data.
You can install mountain lion on all Macs you personally own. To quote Apple: 


  If you've purchased OS X Lion or Mountain Lion from the Mac App Store, you're allowed to install it on all of your personal authorized Macs


In fact, you can bypass installing Snow Leopard at all, it's possible to install Mountain Lion directly: Lifehacker has an article on this explaining how to create a bootable Mountain Lion installer. Here's an AskDifferent answer with a second method.