Re: Rimfire competition question
The smallbore silhouette is done with either irons or scoped, most use a scope, and it is all offhand, 4 different distances.
There is also ARA benchrest, and although there is a "production class", most of the competitors use rail guns, or completely modified alien technology

, to run with them take some $ and some excellent wind reading capability.
This'll get you to the NRA area with tournament information, you can click around and find out more.
http://www.nrahq.org/compete/smallbore.asp
Smallbore competitions:
http://www.nrahq.org/compete/smallbore.asp
Silhouette:
http://www.nrahq.org/compete/silhouette.asp
Nra rulebooks, will tell you what you can compete in with what you have:
http://www.nrahq.org/compete/nra-rule-books.asp
http://www.nrahq.org/compete/rules_images/lite-rifle-folder.pdf
http://www.6mmbr.com/rimfirecmp.html
CMP has a sporter competition:
http://www.odcmp.com/nm/rimfire.htm
Then there is the Ruger Rimfire Challenge:
http://www.ruger.com/micros/rugerRimfire/index.html
Personally, I'd like to see an organized tactical type short and long distance type rimfire competitions, however no one has anything like that as far a nationwide sanction. There are many localized .22 tactical matches or side matches to "main events", but nothing organized as the organizations mentioned above.
I did have a half azzed notion of starting such a thing (heck even NASCAR started from a simple idea) but was put to the side temporarily. I may need to revisit that to see what interest there may be in it. Standardized COFs, a few different rifle divisions, etc.