There is this canister under the hood that has a hose on the bottom of it. The hose follows the red line and connects to "something". Then another part of that hose is just dangling under the car... Where does that go?

 

It's a drain tube, that's how it suppose to be. It's the drain tube (part #7) in the parts diagram.

I also included some other diagrams, even though they don't clearly show a drain tube connected to the canister like the parts diagram.

 

Ah okay. Thank you. Now I have to figure out why my check engine light just came on after getting my stock exhaust wielded up

 

Get the error code. Many places like AutoZone read em for free.
Post the actual code number, not just the guy's explanation (which can be quite creative).

 

Replaced cap today. Had the negative cable disconnected. Check engine light went away, but if it returns, I'll post it.

 

Code 8

8 long check engine flashes, no short

Or Code 80

 

Sounds like you have code 80

Code 80 Exhaust recirculation system

Could you explain in more detail what you had done to the exhaust?

 

The resonator was bad, so they cut it off and put in a straight pipe

 

Well cleaning the EGR ports is where I would start with this one. Most of the times the problem is just clogged ports.

 

267k miles, probably does need cleaning
Never done that before, will be a new learning experience for sure.