Did you hold alt when booting while the dvd was inserted? Just boot while holding alt, it will show you the option to boot from the Install DVD. Then, you can check you Hard Drive (Disk Utility) or perform a Hardware Test.
I'd recommend upgrading to 4GB (if your machine supports 4GB), Mountain Lion with 2GB won't be a joy to work with. If your MacBook doesn't support 4GB it likely not support OS X Mountain Lion, as Apple restricts the new OS to true 64bit processors.
No, the Macbook Logic board won't fit in the MacBook Pro. Maybe you forgot to plug in the battery connector? Happened to me once :) You should also check if the bottom plate is installed correctly (and tight), there are these nasty golden button on the logic board that check if the computer is assembled.
I don't think this is hardware related. Otherwise, Safe Mode wouldn't help. Start normally. Check Console.app for extensive error reports. (Look out for kernel error messages, maybe you'll find the bad kext and are able to disable it, see later in the answer.) Otherwise: Start in Safe Mode. Disable any Login Items (in System Preferences / Accounts) Run the following in Terminal: kextstat | grep -v com.apple This should show you all third party Kernel extensions. You can then disable everything that sounds suspicious by navigating the Finder to /System/Library/Extensions And rename the kexts you investigated to f.ex. ***.kextbackup, then reboot (normally). If the system works, you can re-enable the kexts and login items one by one and find the bad one.