I'm wondering if I should start asking "random" [good] questions, just so I could fill the site with wonderful knowledge for future visitors? Is this more of a Q&A site, or a potential knowledge site (like wikipedia)?
Provided that your questions are on topic and otherwise in compliance with the rules, then I don't think there is any prerequisite for you to not know the answer prior to asking a question. Indeed, I have asked questions which [I felt] I knew the answer to, and others have asked questions which they in turn answered (which is fine).
I would suggest that if you do do so, that:
Make sure your right that they are good questions.
Plan on reading the answers and voting up/choosing good answers.
Don't ask so many at a time that they cannot receive the attention of the current readership.
Yes. Ask away. Just make sure that your question makes sense as a question and that your answer addresses the question.
To your second question: Both. Here's what I have in mind as an unofficial mission statement:
Mi.yodeya matches up people's curiosity about issues of Jewish life and learning with other people's knowledge, getting people's questions answered while creating a public repository of Jewish knowledge, organized in the language of people's Jewish curiosity.
(This is rough and subject to change. In particular, I might work community into there somewhere. Suggestions welcome.)
The main path through which mi.yodeya fulfills the repository part of this mission is via traditional questions and answers, in which the question and answer come from different people. However, if you want to share some knowledge with the world by prompting yourself with your own question, that's fine, too, especially if the question was something you were genuinely curious about before you knew the answer.