Can I install my old OSX on a different computer after upgrading?

Yes. Snow Leopard can be installed on one CPU at a time, not just one specific CPU. If you buy Mountain Lion in the App Store you can install it on all the computers you own.

Is your old SL disk a gray system-specific, or full retail? System specific only works on that one system and you would need to purchase a white retail disk.

To buy retail SL disk: http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard

1) If SL was the original OS installed on that computer, you can't install it on a different computer. The DVDs are tied to the original computer.

2) If you are upgrading ML over SL without erasing the drive, your apps and data will not be touched. Just the OS. However, it's always good practice to backup your system before doing an upgrade.

3) Once you buy ML from the App Store you can install it on as many computers you own or operate without further charges. You just login to the App Store with the AppleID that made the ML purchase, download it and install it on the other computers.

1) If SL was the original OS installed on that computer, you can't install it on a different computer. The DVDs are tied to the original computer.

Technically, if they are tied to anything it would be the computer model, not the physical computer itself. That means you could use it to install OS X on any computer with which those discs are compatible as long as you did not install on more than one computer at a time. I know this is a tad nit-picky, but that's the way it happens to be. No intended offense. Just hoping to be helpful. 🙂

Being accurate, not nitpicky. The OP could be confused if two people with same "builds" reported they did work, while loading the 2009 vintage disks failed to load on a 2010 model.