Thank you for the detailed look at the picture I provided! I haven't messed around with the drone yet since your reply, and certainly interested in trying to clean things up and see where I can get. In the mean time, I wanted to respond and let you know about your observations:

12o'clock position. That's the one I was worried about due to the coloring, but if it's just flux, then great!
2o'clock position. You're right. That is the BEC. I did electricial tape the board itself, but looks like I should've taped the connections themselves. I bet you're right, and that's where the short occured.
7o'clock position. Haha. When I posted this photo, I thought the exact same thing: "Where is that motor wire?" Sure enough, it's just hidden well in the photo, and it IS connected from underneath.
9'oclock. Great to know about the temperature. I think I should probably just invest in one of those soldering practice boards, and give it a go for a while. I have a lot to learn with soldering!

I'll report back after I clean things up a bit, and attempt one or two more power ups!

Solder practice boards are great.

When you go back in to clean things up, take a good look at the other side of your esc in that 2 o’clock position to make sure that that positive cable didn’t obviously damage anything.

If after cleaning everything up and checking for shorts, it won’t power up again- try replacing the bec first.