I am building two lamps from sheet-metal bent to form a globe like
surface. In the finished form, there will be six inner and six
outer pieces that overlap each other. Thus each piece will cover
1/12th (plus overlap) of the surface or 30 degrees (plus overlap)
of the circumference of the globe. It is important that if I look
"directly down" at the edge of one of these pieces, that the edge
looks like a straight line running from the north to the south
pole of the globe.

So, if I were to cut a piece out of the Earth along two longitude
lines (for example 0 and 35W) and lay it out flat (assuming zero
distortion along the edges of the piece - no stretching, pulling,
wrinkling etc.) how could I describe the shape of covered area?
Is it the intersection of two circles and if so, what is the
relationship if their size to the size of the Earth? Or are the
edges of the flat piece described by more complicated,
non-spherical curves just as conic sections result in ellipses,
parabolas and hyperbolas?

A useful answer will supply sufficient supporting material to
give me confidence before I cut the metal as well and also
specify how to scale the answer to fit the real size of my
lamps. A useful answer will not depend on mathematical proofs,
unless you are willing to walk me through it step by step. There
was a time when I could have probably done this myself, but I
haven't used any serious math for 30 years. It's amazing how
much I have forgotten.

Some step-by-step instructions that might help you:

http://octopus.gma.org/surfing/imaging/globe.html

There is no way to do what you ask.  Any flat map formed from a globe
must be distorted by stretching, or have an infinite number of cracks
or pieces in it.  So the shape you would make from 12 pieces would not
be a perfect sphere: if the pieces come together at the poles, any
cross section parallel to the equator will be a dodecagon.
Of course, a dodecagon looks a lot like a circle, so that may not
bother you.