You don't need studded tires in Brooklyn. The streets are plowed and salted pretty well and there are not a lot of ice issues. I don't ride in storms, but otherwise I ride most of the time. I don't know anything about gluing the tire to the rim. That doesn't make any sense to me. It'll sure make it pretty hard to fix a flat.

Yes, take your bike inside at home. Please. If not, you will probably be posting a sad thread soon about how your bike was stolen. Also, there is a new law in New York about allowing bikes inside office buildings. You should look it up and see if it applies. Don't lock it outside unattended unless you really have to.

Kent and Flushing is the best route, probably. I find Flushing to be generally pretty clean and fast.

Clothes you will have to experiment with, but keep in mind that you won't be as cold as if you were walking due to the physical exertion.