In answer to question 1: Some motorcycle engines are well-balanced, so they vibrate very little. Others, such as those in Harley-Davidson motorcycles, are inherently unbalanced, due to their 60° V-Twin engine design. Many of these have vibration dampers on the sprocket shaft of the engine, the shaft that drives the primary chain. Some of the older H-D engines were directly mounted to the frame, so engine vibration was transmitted to the rider. Starting in about the 80's engines and transmissions were rubber mounted, so riders didn't feel the vibration as much.
I'm not familiar with what other motorcycle manufacturers do.
For question 2, car engines are connected more directly to the transmission, rather than use a primary chain. For a standard transmission, the clutch with its pressure plate is attached directly to the flywheel, so it wouldn't be feasible to mount a vibration damper between the engine and clutch. For automatic transmissions, the torque converter isn't a direct connection between the engine and transmission, so engine vibration isn't tranferred to the transmission.