Question 1 What type of nails do I use for the real wood crown molding?
Question 2 How do I measure the angle, in degrees, of my ceiling? It is something other than 90 or 45? I was hoping to use something simpler than Trig functions.

I use an airgun for crown moulding.

As for the angle, it would be 45 degrees - unless vaulted.

The Nails U Need To You Should Be No Shorter Than 2 1/2 Air Gun Nails,
A Standard Inside And Outside Miter Are 45 Cuts.
Remember You Have To Cut Crown Upside Down And Backwards, And Bed It In The Mitersaw Correctly, Good Luck Nichols Trim

Thanks for people who are trying to help by posting an answer to the question.

I hope that people posting an answer to this have noticed the date on the original post.

I would think that the original poster has probably already found an answer to their question.