A couple of friends and I would like to establish a non-profit
corporation for the purpose of creating software and managing a
network running our software and others. Further, we would like to
have donations to our organization be tax deductable for US-based
donators. So, I will break this down into parts:

1) Under what category/purpose can we form a non profit corporation?
2) I am a resident of Arizona, but the people on the team live in
other states and even other countries. What state should I form my
corporation in, in order to minimize costs. Also note that we have
only 2 founders who are based in the USA. The other 2 are in Germany
and Australia.
3) Someone may have attempted to "beat us to the punch" and registered
the corporation in California, but I do not find the listing in their
database, so I don't know if this person is lying or not. How
up-to-date is the information in California's online corporation
database (http://kepler.ss.ca.gov/list.html), and how can I find out
if this other individual has already started the registration process
for our corporation name?

Thanks in advance.

Generally, non-profit organizations are formed to serve a specific
service of benefit to the community - not, as in your example, to
develop software.

You might check http://www.state.ma.us/sec/cor/CorpWeb/cornpc/npcinf.htm
as an example of allowable purposes for a nonprofit corporations.

Technically, this company would develop free software for the
betterment of a community of players who play a certain online
computer game. Supposedly in California, we would be able to file as a
"non-profit mutual benefit corporation". I hope this helps clarify my
question. Thanks. :)