I'm not sure what Alexei is using to record his videos, I think it might be a digital camera that takes video too. I know he uses a tripod held up with bungee cords, seems to work alright for rallycross but does move around a fair bit.
The setup I have is a bullet camera from
http://www.chasecam.com , their 520 line version with audio.

This camera feeds into my miniDV camera which I just use as a VCR, basically. Chasecam also sells a nice digital solidstate recorder that uses memory cards, which if I didn't already have the video camera I'd be using.

For mounting, I use their triple suction cup mount, which works great for rallycross stuff (as long as you make sure the surface you're attaching it to is perfectly clean, it has come loose when I've been careless and not cleaned the dust off). I don't think I'd trust it outside the car for stage use like I've done for rallycross, but it works well inside. They also sell lots of different mounts for rollcages, etc.

As you can see in the following videos, the suction cup mount is very stable, even in a stage car on a rough road. This is mounted to the windshield.
http://media.putfile.com/Mark-Williams-Test-Day---Run1http://media.putfile.com/Mark-Williams-Test-Day---Run-3http://media.putfile.com/Defi-mulgraveHope that's helped. Of course, lots of people use normal small videocameras as well, on rollcage mounts or the like, with varying degrees of success.