Does MythTV offer the option to automatically record programs it 'thinks' I would like? This being similar to TiVo's automated recording functionality with the whole thumbs-up and thumbs-down scheme?

I think you can program it to record shows with certain keywords like "Nature" for example.

Unfortunately, MythTV does not have a feature like TiVo's suggestions. I miss this feature.

There is a MythCollaborative plug-in:

http://www.mythtv.org/wiki/index.php/MythCollaborative

No idea how well it works or how many people use it; obviously the recommendations aren't going to work well unless a good number of people are using the plug-in and providing data.