I would like to be able to search for multiple URLs that contain a
unique word (i.e., illinois), or, to search for all URLs that begin
with a certain letter/alpha combination and to be able to make these
requests against a global database or have the cabability to choose
multiple domains (.com, .us, .ca, .es, .whatever, etc.). In the past
there were a few sites that offered this type of search capability but
they seem to have disappeared. If these types of search engines are
avabilable I would appreciate any and all examples you find.

Have you tried http://www.searchthingy.com/search.asp ?  It searches
multiple websites at one time.

Try TurboScout.com and see if that is what you are looking for.

http://www.allsearchengines.com/ i strongly recommend this site! there
are all the best search engines that i know, so i hope u find what you
are looking for^^

I'm surprised that no researchers have answered this yet, and all 3
commenters have misunderstood your question! A couple of sites that do
exactly what you're looking for are:
  www.namedroppers.com. 
  http://www.whois.sc/domain-explorer/

These, however, don't search *all* domains, but do search the most
common ones. A good researcher should be able to give you some more
links to similar sites, but hopefully this suffices for now.

Google search strategy:
*  search domain name "starts with" contains
*  domain search

pianoboy77a-ga - thanks for the comment. You're right,
namedroppers.com and whois.sc are both the type of search engines I am
looking for.   Someone had previously suggested whois.sc but it looks
like the comment had been canceled.
Thanks Much