I have a raspberry pi 3 (that I'm using to ask this question) and I want to know if it is possible to download the full version of the game onto it. I'm running the Raspbian Debian-based OS. On the side, would I be able to download mods from other people onto it?
The short answer is no.
The raspberry pi's hardware just isn't up to the task of running minecraft, much less minecraft with mods. The version already installed on the pi has been cut down and optimized so that the pi can handle it. If you compare the specs of the raspberry pi 3 and the minimum requirements needed for minecraft, you will see that minecraft needs almost double what the pi can handle.
This became practical with the advent of the Raspberry Pi 3 in February 2016 (hence your question), improvements to Raspbian OpenGL drivers, and tenacious tinkerers like rpiMike.
- How to setup Minecraft 1.8.9 on RaspPi2 (original tutorial with Minecraft 1.8.9)
- How to setup Minecraft 1.12.1 on Raspberry Pi 3 (updated for Minecraft 1.12.1, OptiFine, Raspbian 9 Stretch)
Both threads contain links to tutorial/demo videos. This gives you an idea of the range of performance you can achieve. I can attest that on Stretch the quick setup which tends to work out of the box can get you a stuttering engine that renders at about 10 fps. If you put in time to enable OptiFine and go as far as overclocking, apparently it is possible to achieve 40 fps with almost full screen resolution.
On the side, would I be able to download mods from other people onto it?
This is more difficult at the time of writing. If you read toward the fifth page of the v1.12.1 thread, you'll see one user claims to have gotten Forge working while others are having trouble. This setup will likely get cleaner over time, and if you're up for debugging Java issues you could troubleshoot and support it yourself.