The easiest way is to use a plastic tub with a lid. You can get one for $5 at Walmart
I wouldn't worry too much about the solution cooling off. As long as the water temp is between 75-85 degrees you shouldn't have ay issues. (think water temp that you would bathe a baby in)
A single 75 - 100 watt Infrared heat bulb should do the trick unless your room is really cold. Infrared bulbs can be used 24/7. I like this particular bulb: http://www.petsmart.com/product/inde...AvailInUS%2FNo
The only place I've purchased betadine before is Walgreens, so I can't really give you a recommendation on that one. a google search is probably your best bet unles someone else comes along.