Can I transfer pre-installed apps from a device with a broken screen to a different one?
You can't transfer system apps (that is, the apps that came preinstalled) to another device or back them up. The apps will be built into the Funtab's ROM (or system image), and even if you could extract the files, they almost certainly will have copy protection to stop you using them on a different device.

To make an analogy, imagine you bought a PC and it came with PowerDVD. Now imagine you were to break the PC and buy a completely different one from a different manufacturer. You wouldn't have an install disk for PowerDVD, and the old installation would have copy protection so you can only use it on that specific model of PC. If you wanted to install PowerDVD on your new PC, you'd have to go out and buy it. Preinstalled apps on Android are no different.

If you'd downloaded Angry Birds from Google Play (or most other marketplaces), you'd be able to install it on any device with your account. You wouldn't need to back it up: Google would know that you'd bought it, so they would let you download it again for free. Preinstalled apps don't work that way, though.