What are methods used to cool a small residential enclosure (cabinet)?
Formalizing my answer from the comments section: Look to see if you can (passively or actively) vent your cabinet via the bottom into the crawlspace under your house, since that is not visible but still accessible. Make sure you don't overcomplicate it though, and make sure you don't damage anything structural in your house (e.g. don't cut through any joists :) and watch out for existing ductwork and wiring).

Oh, and remember that heat rises, so you will likely need to use the active venting route with a fan, but since you have access to power for your server and AV equipment, I don't think that will be a problem ...