Any suggestions for places to stay in Mandalay? My main requirements are clean, no bedbugs, and under $20usd per night.

Any place that you can rent a bike would be great too.

The Golden Mandalay Hotel is a nice place to stay. They have a web site so just look it up. I paid $20US for a nice big double room (for two), however, it was peak tourist season and hard to find a room so it is probably much cheaper now. The family who run it are lovely and will help you out with any information you need.

You will not have a problem finding a guesthouse in Mandalay. But be prepared to pay 10 to 15 US Dollar for an air conditioned room with bed and shower.

Mandalay is a little bit cheaper than Rangoon but still quite expensive in terms of accomodation. Food and Taxis are much cheaper than in Rangoon with food beeing cheaper and better tasting than Kunming and Taxis beeing much more expensive (no taxometer) and often borderline scrapheap.
Don`t visit Mandalay castle unless you get really bored. It is 10 Dollar and really not worth going inside.