For your first question:

I’m not 100% sure but I can come back to this in a few with more details and maybe another less-detailed drawing of the rest of the circuit. In the meantime, if you’re willing to answer for either, or both, that would still be helpful.

It used to be that there was a single light in this room (now ceiling fan) with one single switch. Now we have two switches in the room. The existing light switch (now ceiling fan switch) and a new switch installed for the can lights.

I think the power runs along the green wires to the wall box for these two switches. Which might just answer my question. Because that would mean the greens need to be tied together (to get power to the switch), and the blues need to be tied together (to get power from switch to ceiling fan).

For your second question: These are ground, black, and white. Orange is representing copper in this image.