Would it be possible to install the T3i software on my T5 (1200D)? Both cameras have the same Digic-4 CPU. The reason I'm asking this is that the T3i have some features for video my T5 doesn't have.
No, T3i firmware will not install on your T5. Firmware is always very specific to each camera model.
Magic Lantern is a free 3rd party firmware add-on that is available for many different Canon DSLRs. It unlocks many hidden features, but it is not yet fully developed for the T5/1200D.