I have owned nothing but air cooled bikes my whole life and from my experience, it isn't uncommon for them to vibrate or act weird when cold. As far as the cam chain tensioner, it is automatic on the Bandits and don't require attention unless it is broke. If it goes away when warm, it most likely isn't the the motor mounts, but doesn't hurt to check. Bandit motors are typically known to be vibey motor, especially around 4K rpms. I've had motor mounts come loose, so it doesn't hurt to check them periodically. I do notice that the bike is a bit more determined not to run right in the Winter than during the Summer months until the motor is warmed up. Carbs don't compensate as well as fuel injection does. What kind of oil are you using and what weight?