The only potential issue with using an external drive for images of the OS drive is that sometimes actually restoring that image can be problematic. It requires booting to a rescue disc and being able to access and restore from an external drive. I would want to actually perform this operation to confirm that it can be done on your motherboard. For the best solution, a second internal drive is optimal for image storage, since it can be easily accessed from rescue utilities.
Backup software that simply copies data to the external and runs in Windows is another option, but does not eliminate the need for a complete re-install of Windows and all applications if your main drive fails.
No, I would not install programs to an external drive. Simply because the programs will still install files and registry keys to the system drive, so again you would be re-installing the program in the event of a system disc failure.