Should a teabag or looseleaf tea brew at room to cold temperature water? I recently noticed that an organic bigelow tea(teabag) starts brewing at room/cold temperature water. Not sure if it is brewing or not but the water starts to get green after dipping the tea few times. 

I seriously doubt that tea could brew in cold water and that too in few dips. This has put a doub in my mind, whether I am drinking tea or something flavored? The ingredients on the packets say: Organic Green tea and nothing else!

Should I stop drinking the tea from tea bag and buy loose leaf teas only? Any suggestions/tips are appretiated.
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.