Call the Writers Guild. I and my business partners have all the details and the concept is solid and saleable. I am not a game show expert. However I suggest that you put down on paper your concept and try to contact the few production companies that produce gameshows - i.e. King World, Mark Goodson Productions, etc. It is very difficult to sell game show ideas to established producers and even harder to the networks.
Short subjects can be great writing samples.
In the feature market it doesn't mean much. However they are good to open doors, if its writing quality is excellent. Regarding your second question, the answer is yes. Submit to agents that accept unsolicited material. We have their list in our Holywood Access Directory.