When is the next event?

Well, unless there's one this weekend that I don't know about (very possible as I haven't been looking for one), the next one could be the SunRiders autocross on March 9th.

How to prepare?

Why do you need duct tape for your tires??? Definitely come prepared for the typical outdoor event. (water, sunscreen, yadda yadda) Don't worry about what other cars will be there, it doesn't matter, especially not your first time. An AWD DSM is a fun autox ride, though maybe not the ideal one.

Check out the autox novice guide on the Tire Rack or do a web search for "autocross novice guide", you'll come up with several.

Local autocrosses are pretty casual and I've never heard of any giving cash prizes, though most do give out trophies for the class winners. If you stick with the sport long enough and want to do larger National level events such as SCCA National Tours and Pro Solos, then there is money to be won, mostly in the form of contingencies from sponsors. But you're a long way from having to concern yourself with any of that. Just go have some fun, feel out "the game" and see if it's for you.