The camera hooks into your laptop. If you have an HDMI port on your computer (or the proper converter to HDMI) you can mirror your laptop on your HDTV, with or without the camera. 

You can control camera shake by using your mirror lock, camera shutter release and cable, tripod securely tightened and all legs as squat as possible. The higher the tripod the less stability you have, but you can always attach your camera bag to hang on the hook, located on the bottom of the tripod to act as a center weight. And lastly, if you do not have a camera shutter release cable, the higher your Shutter Speed the more you will be able to control the “camera shake” or blur in your photos. I hope this helps. 

Yes 

In my case I did clean the mirror with cleaner and it worked fine. My camera does have a focusing screen, because of the design, and I would be very careful with that screen. This is based on other older film cameras, I would just use a blower or a fine brush. 

Cheers.