I'm just playing my first game now, and I'm wondering how some elements will play out.

First of all, is Funding public knowledge, or private. Can I fund someone without others knowing?

Second, what happens when a player's queen is captured? There is a distinct lack of information on that in the rulebook. What happens to their outposts? Do they go dormant? What about their drillers? Are they all destroyed, or do they just sit there?

1) any player can see how people are funding a specific player, but only the funder and the person being funded know who did it. Funding is anonymous to every except the played involved.

2) if a players queen is captured/assassinated/hypnotized that player is eliminated

Thanks!

To clarify number two, a player is indeed eliminated when they lose their queen (unless they have a princess, in which case, the princess is promoted to queen).
Their outposts remain in their possession, and their specialists remain active, but the player can no longer influence them. This means the following:
1. Any scheduled orders are immediately cancelled.
2. Any active movements continue on their course, and play out normally.
3. The global effects of any of their specialists remain in effect (so a security chief will still increase all of the ex-player's shields, for example).
4. So do the local effects (so an eliminated player's sentry will still fire on you, for example).
5. The factories continue to produce drillers per the standard rules.
6. The eliminated player can still see anything within their outposts' sonar range.
7. The eliminated player can still interact with other players via messaging.