Yep.
The main mirror in a D5500 is up out of the light path blocking light from the viewfinder. So the light can get to the image sensor and be displayed on the rear LCD..
Use a gimbal to steady the cam. You can even make it yourself with a small power supply and 3 motorized gyros, one for each direction of dampening. An cheaper alternative is to modify one made for a smaller camera like a gopro.
As an alternative, use a tripod, or make or buy a handle mount.
Yep. None of those Sony camera actually have a "viewfinder".
Those are SLT cameras, not DSLR cameras, and have a electronic viewfinder (EVF) - a miniature electronic display in effect showing what a DSLR shows on the rear LCD when in live view mode.