I am interested in the regional use and possibly the origin of the
term "underdog". When children push each other on swings they often
run under the person on the swing and call this an "underdog" Is this
only used in cetain regions of the country. How common is the usage?

It appears that the word was coined in 1887; while there doesn't seem
to be much concrete evidence or "story" behind the origin of the word
available, it is quoted as originating in 1887 in the Merriam-Webster
Collegiate Dictionary.

I believe this term to be in common use, at least throughout English
speaking countries, of course; I've heard it on any number of
television shows (primarily sports events, of course, where the term
is used more often than in an everyday conversation.) The cartoon
character mentioned in the comments is more likely to have originated
the "playground" usage of the word, but the cartoon character was in
turn named after a term created over a century ago as.. well, a
synonym for "loser." :)

(As for how commonly the "underdog" thing is on elementary school
playgrounds.. I remember the term myself. I imagine it was quite
common among Generation X kids, but I imagine usage has dropped
drastically with the advent of newer, louder, more obnoxious kids'
fads. Pokemon, and what-have-you. :)

Hope this helps!

-brightshadow

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From Princeton University's "WordNet" (parsed by dict.org):

  underdog
       n : one at a disadvantage and expected to lose

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From an "Online Etymology Dictionary":
[Sources quoted on http://www.geocities.com/etymonline/ for this
listing of words]

http://www.geocities.com/etymonline/u1etym.htm

underdog - "the beaten dog in a fight," 1887. 

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From Merriam-Webster's Collegiate Dictionary:

http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=underdog+

One entry found for underdog.
 
Main Entry: un·der·dog 
Pronunciation: '&n-d&r-"dog
Function: noun
Date: 1887
1 : a loser or predicted loser in a struggle or contest
2 : a victim of injustice or persecution 

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Search terms used:
etymology underdog [google]
etymology [dict.org]

This is purely anecdotal, but it may inspire someone else to go a bit
deeper into the spread of its usage.
I grew up in northern CA and we frequently preformed the underdog
manuver on the swings. On my school playground, it was done as an
imitation of the cartoon character Underdog. Underdog ran on saturday
mornings on NBC from the mid '60 to the early '70s and was esentially
a cartoon dog version of Superman. Underdog frequently knocked out
villian by running at them whith his fist out, and striking them so
hard they flew over his head. I believe imitation of this as well as
the "pun" of running "under" the swing inspired the name of this move.
I am quite surpised to hear the term is still being used.