There are a dozen shops south of Paapu House. Usually you have to leave about 2000 THB deposit plus copy of Passport. A 125 cc bike like Honda Click, Yamaha GT will be around 200THB per day, sometimes more expensive depending on the time of year. BAMBOO BIKES, 200 m down the road, is good, no nonsense, although sometimes old bikes. Tell them you come from Paapu House, you will get some discount.
INTERNATIONAL DRIVING PERMIT - business model of the local Chiang Mai Police is pulling over tourists on their bikes and asking for an international driving permit, that is a grey booklet, and it should contain an A-stamp for motorbike (125cc feels like a moped, but officially is a motor). They will have checkpoints all over town, mostly on the moat, on the corner close to Paapu house, on the way to Doi Suthep, on the Nawarat Bridge, daily. You will be charged 500 THB if you are not able to show the IDP. You will have to pay at the Police station within 24 hours, do not pay in the street if you do not get a ticket: the fine ticket will remain 'valid' for a few days, if they pull you over tomorrow again you can say 'i already have a ticket...'
Wear a helmet anyway, fine for no helmet is 2000 THB.