Camera-IMU (Inertial Measurement Unit) sensor fusion has been extensively studied in recent decades. Numerous observability analysis and fusion schemes for motion estimation with self-calibration have been presented. However, it has been uncertain whether both camera and IMU intrinsic parameters are observable under general motion. To answer this question, by using the Lie derivatives, we first prove that for a rolling shutter (RS) camera-IMU system, all intrinsic and extrinsic parameters, camera time offset, and readout time of the RS camera, are observable with an unknown landmark. To our knowledge, we are the first to present such a proof. Next, to validate this analysis and to solve the drift issue of a structureless filter during standstills, we develop a Keyframe-based Sliding Window Filter (KSWF) for odometry and self-calibration, which works with a monocular RS camera or stereo RS cameras. Though the keyframe concept is widely used in vision-based sensor fusion, to our knowledge, KSWF is the first of its kind to support self-calibration. Our simulation and real data tests have validated that it is possible to fully calibrate the camera-IMU system using observations of opportunistic landmarks under diverse motion. Real data tests confirmed previous allusions that keeping landmarks in the state vector can remedy the drift in standstill, and showed that the keyframe-based scheme is an alternative solution.