I need to find a publication that explains the different temperatures
based on the different depths of a paper hot cup holding hot coffee.

For example, in an 8 oz. cup of coffee in which the cup is
approximately 4 inches tall, what are the differences in temperature
1.) at the very bottom of the cup, 2.) at the 1/2" mark, 3.) the 2"
mark, 4.) 2 1/2" mark, 5.) 3" mark, 6.) 3 1/2" mark, etc.

Someone once showed me these differences when he used a special
computer monitoring program. He had a measuring device that had
different temperature cables with nodules emerging from it (which
hooked into the computer). He placed 6 nodules into a freshly poured
cup of coffee at precise heights/depths. He then had the computer
measure the difference of each of the 6 nodules over a period of 5
minutes. I no longer have this data, but would like to find a
published report if possible.

I need the information to come from a trusted source so that I can
quote the publication.

Through the use of a computer model and using engineering principles
more info is needed.  What is the diamaters of the cup (Top and
bottom), the thickness of the paper, the orginal temperature of the
coffee, and the outside temperature to name a few?  Can we assume that
there is no wind, no other heat sources in the room, no lid, and plain
black coffee to name a few.  Are you trying to see how the
temperatures change from the pouring to the temperatures stay
constant, the rate of cooling, or the different temperatures at
different levels?  The temperature gradient will be very small but I
am assuming that the temperatures on top wil be higher than the ones
at the bottom.  I hope this helps.

I don't have specific measurements for a cup, yet the idea in mind is
for a traditional 8 oz. paper cup, e.g. a Starbucks 8 oz. cup with a
plastic lid. Yes, you can assume no wind and plain black coffee. My
goal is to find data that enables me to make a conclusion about the
temperature of coffee from the top, middle and bottom of the cup when
coffee is poured at 180 degrees Fahrenheit and eventually cools down
to 100 degrees Fahrenheit. From the one test for which I don't have
data for, I believe the temperatures at the bottom were hotter than at
the top.

I would like to be able to reach this conclusion based on data from a
published article:

"When coffee is poured into an 8 oz. paper cup with a starting
temperature of 180 degrees Fahrenheit, the temperature at the top,
middle and bottom of the cup vary only by +/- x.xxxx degrees
Fahrenheit throughout the entire cooling process (when the cooling
process is defined as 180 reducing down to 100 degrees Fahrenehit)
when a plastic lid is placed on top (and vary only by +/- x.xxxx
degrees Fahrenheit when a plastic lid is not placed on top)."

I did find an article that asks the precise question that I need data
for: "What if we wish to know the temperature at any position within
the cup of coffee at any particular time?" Yet, the research performed
center on ice-cold water because it was easier to work with. So, one
of the items in the section "Further Questions" includes, "How well
does this model work with hot coffee in the thermos instead of cold
water?"

Modeling Heat Flow in a Thermos. American Journal of Physics, July
2003. With J. Scherschel.
http://www.bsu.edu/web/mkarls/

Using an equation for 3D heat convection, you could probably do this. 
Unless the results are already available, this is most likely going to
cost you much more than $2.50 for someone to model this particular
situation.