Is there a way to get the full version of Minecraft on a Raspberry Pi?

Yes.

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.

For reference:

• 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.