that's a lot of questions.

for nails, i would use double hot dipped galvanized nails of
appropriate length. as a rule, nails should be twice as long as the
thickness of what you are nailing, and should penetrate the underlying
wood by a minimum of an inch. stainless steel is overkill for a
woodshed. the only place i would use them is if anything is nailed and
sitting on or below the ground. not sure what you mean by coated
nails.

for fastening the 4 bys, i would use the star drive lags. otherwise
3/8" lags long enough to get a good bite should be fine.

you can still get .60 treated, though i'm not sure if it CCA or what.
i would use for your skids.

if you have lightly treated wood (look at the endgrain of a cut and see
how deep the treatment goes) treating the endgrain may make it last
longer. but that step is typically not done by modern carpenters. so
it's not essential, no.