You could use the VGA (second monitor port) connection to run from the laptop to a LCD TV if it has a VGA input. This would just give you a second monitor for the laptop that you could play video on. Audio would have to be connected separately.

The lens is what helps the camera shoot at higher shutter speeds. Any lens will let a camera shoot at 1/4000...just a faster lens will let in more light.
Without knowing what focal length your looking to do and if 1/4000 is the goal speed for a good exposure look for a f/1.2 lens.
You could get by with f/1.8 for some shots with a lot of light.

All Digital-SLR cameras have fast enough shutter speeds to do this shot. Be it a entry level or pro-range camera.
What helps is having what is referred to as a "Fast Lens". The reason it is called "Fast" is that the aperture will go to a large opening to let in a lot of light to allow the high...fast... shutter speeds. The numbers are a little confusing in that the smaller the number the larger the lens aperture will go.
A f/1.4 is faster than a f/1.7...and that is faster than a f/2.8. The smaller the number, the bigger the opening and the more light is let in..the more light the higher the speed of shutter that can be used.
If you don't have a fast lens it wouldn't matter what the max shutter speed the camera has because there wouldn't be enough light to allow exposure.

It should be able to read and access the card but will probably only read at up to 40 mb/s peek.