What are these and what do they do? I'm not sure, but I think it may be a thing from the Rise of Mankind mod. Thanks in advance. :)

Well I don't play RoM. But the Trade Mission I know about is when you get a Merchant GP. Your options for the Merchant - depending on the Era your game has progressed to:
Join City / Research a Tech / Initiate Golden Age, Trade Mission (and in BTS: Found a Corporation)

If you select the Merchant and hold SHIFT, move the mouse over to some other foreign CIV's city - the TradeMission icon will light up (Keep holding Shift) - you then hover your mouse over the TradeMission icon to see how much that particular one will fetch you.
Now hit CancelMove, and repeat the process to check out some other Foreign cities - and pick the one you think is best :-)