We were slightly worried about it too. The reason we chose to be generic is that we want to be the de-facto app to solve needs. We're trying to keep it narrow by the kind of communities we market it to.
I like what you're saying about keeping it narrow by the communities you're targetting. If it's truly useful, and generic, then people will start finding their own alternative uses.
I wouldn't have three different type of question on your front-page header;
1. Lost my Dog, if you spot it ... 2. Wanna play a game? 3. What is...
I'd pick one type (personally, the 2nd) and stick to that. 1. is only useful with significant localised usage and 3. is better left to something like twitter for now.
Whatever you do, best of luck.