Do you know of a video editing application that is as good for video as Audacity is for audio?

I love Audacity. It is easy to use and can do a lot of things, and it’s open source!
I’m interested in a similar application for editing video. There are two features that I’m specifically looking for:
1) Frame-by-frame advance.
2) The ability to turn any particular frame into a still photo.

I’d like to explain #2 a bit more. While taking a video with my camera, if I see something that I want a photo of, I press the shutter and get a still photo in addition to the video without affecting the video. I’d like to be able to do the same thing with an existing video.

Can anyone recommend an open-source (or low cost) application?