Convenience Stores
Chiang Mai has its share of Family Mart convenience stores and 7 Elevens. They're fairly prevelant due to the high traffic of tourists so you won't find it difficult to find one. There's a 7 Eleven next door to the Raming Lodge on Loi Kroh Road.
Internet
You can hardly find cheaper internet access than in Chiang Mai. On Soi 1 off Loi Kroh past the Number 1 Bar, there is an internet cafe that charges 30 Baht for 1 hour of internet use. That amounts to less than a dollar for one hour of internet service. Wow! On main roads, you'll pay a little more but internet access in Chiang Mai is incredibly cheap.
Laundry
Your clothes can get washed and dried very cheaply in Chiang Mai. But note that many smaller places hang your clothes to dry under a tarp. You may want to stick with hotel laundry service.