I once spent a month in that region from early February to March. That seemed like the perfect time to be there, very warm but not oppressively hot (around April/May it could be) and no rain. Actually I've never been there during the rainy season, so can't comment on that.
I rented from my friend near Bangkok, and once I took my own bike there from Europe, so don't really know about rental options in Chiang Mai. But there are several of them. www gt-rider dot com should be the website that has the info. Plenty of route options to be found there as well. But all around Chiang Mai it's a rider's paradise, simple as that! The very best roads for me have been close to Lao or Burmese borders. The so-called 'Mae Hong Son loop' was fantastic, but there's no shortage of equally great options.