New Roof - Exposed Nail Heads
There might only be exposed nail heads at the end(s) of the ridge.  The other nail heads should be covered by the adjacent ridge cap shingle, like this (from here):



A professional would hide the last nail heads by gluing an end piece on like this:


That seems like what you are describing.  If so, there should be no nail heads exposed.

If you paid for the roof, have the guys come back and fix it at their expense.

As for drip edge, that might vary according to local climate and neighborhood restrictions.  (I used to live in a neighborhood which required cedar shake roofs with non-metallic edges.  The HOA successfully sued a homeowner who did otherwise.)  Check with your neighbors.