In accordance with Thai law, everyone is required to carry a form of identification with them at all times.

Make a photocopy of your passport and the page with your visa stamp. A picture of your passport, shot from your phone, will work as well. Always keep your passport or the photocopy with you (the law requires that you carry your actual passport at all times, but in practice a photocopy will usually suffice). Many night clubs insist on a passport (and ONLY a passport) as proof of age. It is not required that you leave your passport with a hotel when you check in.

Motorcycle rental places are likely to ask for you passport as collateral. Leaving another document (drivers license, int. drivers licence, second passport, ID card) might be the better option. Or consider offering a cash deposit instead.