This is literally my second time even getting my squad on the battlefield, so I don't really know how to do anything there. 

I'm trying to get Thallo; I need a nightcat. I know it's possible to get one from the portal zone. What I'm wondering is if you pick the item you get from the battlefield or if it's random; if there's a gene that let's you pick, etc etc.

Basically what can I do to maximize my probability of getting a Nightcat on my own?

You get to the portal (I've managed to twice, yay) and then you wait all day until there's a "trade request from one of the moons", you click the little button that says trade and then it tells you you succuessfully completed the trade, click here to see the item that you got in return.

I suspect that right now they aren't random, but only nightcats, to keep up with demand :)

I would hope that they are only nightcats, considering the event. Anyway, good luck to you!

lol, there is only 1 trade request per day, so it's better to take the portal zone before the time change / right after the change.

As for picking items, it's random like the chests, though you can increase the chance of what you're looking for appearing with the Astrologist.