1. Should any of the tub drain/overflow be attached to the tub before installing it, or do you prop it up on some type of skid (2x8's on edge?) and then attach and align the drains, and then drop it into place?
It depends on the situation. The easiest way is to assemble everything and then set it into the drain pipe. But more often, either the drain has to be connected to the drain and then the tub placed over it, or the tub is set into position and then the drain assembled to fit it.
2. The tub drain has rubber gasket. Do I need some type of sealant between then gasket and the tub and/or drain pipe? Silicone or some type of putty? Should I use silicone on the top side of the tub between the strainer basket and the tub?
Nothing between the tub and the gasket. Putty, or silicone if you prefer, between the strainer and the tub.
3. Are there standard distances between the top of the tub and the spout? Are there standard distances for the shower head?
No to both, just as there is no standard distance for the valve above the tub.
4. The tub installations says nothing about ledger boards. Should I install them any way? If I install them, how do I get a good fit? Can I go by the dimensions of the tub and leave a little extra margin, or does it need to be dry installed in place, marked, then move the tub so I have enough room to nail, then move the tub back so I get goo support?
You should use them so that the tub rim cannot flex and create a leak point. Measure the tub, or use the installation drawing if you have one.
5. I'm going to do a mortar bed installation. Is the mortar mix you can get at Lowes or Home Depot OK?
What you use is immaterial as long as it will harden.