How does Picasa decide how to process RAW photos? When I shoot in RAW+JPEG the two pictures can look dramatically different. Is it just converting the RAW files upon import, or is post-processing still possible in the application?
I don't know all the technical details of how Picasa handles RAW files, but my short answer would be: "Dont even bother". There's a good blog post on using Picasa with RAW files.
Don't get me wrong. There is nothing wrong with Picasa. It's very good at what it's made for. I use it all the time for basic JPEG editing and uploading baby snapshots to share with family, but when I need to process some RAW files I'll always go to Capture NX or Lightroom (one day when I figure out how to handle colors in Nikon's RAW files).
First of all, it sounds like you are comparing Picasa with your camera's JPEG generator, so I'll talk about what your camera does first, and the I'll talk more about how Picasa works.
A typical camera will take an image on the sensor, and it'll be some kind of a RAW image. As it takes an insane amount of time to write an image out, most of the time the conversion to JPEG is done using a specialty chip that handles the conversion. They typically have a few settings, like sharpness, contrast, white balance, etc, but the control is fairly minimal, and you're stuck with whatever comes out the other end.
Picasa will do some preliminary processing on all RAW images upon import. Picasa, like most serious photo editors, doesn't actually change the RAW image directly, instead, it contains a sidecar file that lists the changes that have been done. There are a few exceptions with Picasa, namely when you do spot removal, or red-eye removal, a JPEG is created, but otherwise, the only changes occur in a sidecar file.
What Picasa does out of the box isn't completely clear, but it is clear that it does some kind of auto contrasting. I've noticed that when I've used it, the highlights are almost always over exposed, where in camera or using Lightroom, I don't see the same problem. Of course, you can fiddle some with the dials to try and improve the performance, with limited success.
Picasa, for it's price, is a great program, but like most things, you get what you pay for. I do recommend it as a whole to a beginning photographer, but if you're the type to spend hundreds on a camera or lens, you should consider a few hundred to get a better post processing program.