that will be fine.. u can compost in the winter too, it doesnt stop things from decaying only slows them down..
Keep it off the concrete floor via whatever works. Add a cup or two of alfalfa meal, moisten, cover, LITFA for a week, and when you take cover off i can promise you'll notice that things have indeed been cooking as moisture will collect on the inside of the cover. But the most direct answer to your OP is to elevate or insulate from direct concrete. Cold concrete sucks the heat outta just about everything it comes in contact with. Check it out....
IDK about that as a necessity. Reads like you're just trying to get some mineralization going with the cycling processes. IDK much about worm farming except to say this, if the worms do not like their environment for any reason they'll do their best to escape from it and in doing so are willing to die in the process. Like lemmings off the edge of a cliff. That's my personal experience anyway. So, IDK if you need worms per se, what you actually need is the finished product, EWC. Worms wont stay in a worm farm hotel unless they get the 5 star treatment all the time. Keep It Simple Sir
IMO if your cycling/cooking your soil you must have a high N source like alfalfa added. If not you probably don't need to 'cook' it. If so, don't add worms till it's a month old. I just mixed more soil with alafalfa so at a month I'm gonna introduce worms. Good luck