Hi John,

I'm using 1/2" drywall
Does it matter if I use 1 1/4 or 1 5/8 screws?

The screws should penetrate at least 3/4" into the stud, so either screw
length would work fine in your situation. Use whatever is cheapest and/or
most available.

However, be sure to install steel protection plates anywhere wires or pipes
are less than an inch or so from the face of the stud. Otherwise, a longer
screw could theoretically go deep enough to pierce the wire or pipe.

when hanging the sheets do I have to install all the
screws in each sheet before moving to the next

I don't know how many screws you would have to install to hold the sheet
temporarily, but why? It's better to finish the sheet and move on to the
next so you don't forget anything. It takes the same amount of labor either
way, so finish the first job before you start a new one.

Anthony