Should a teabag or looseleaf tea brew at room to cold temperature water?
Yes, tea will brew at cold temperatures.  Growing up, we'd make sun tea by leaving it in a sunny window for about 8 hrs.  These days, I put it in the fridge to brew simply because you have a longer window before the more bitter flavors start getting extracted.

Tea bags will brew faster, as they tend to have smaller pieces and possibly dust in them, which means increased surface area.