Have a look at the following simplified snapshot of my current MusicBee window:
http://www.picfront.org/d/9hwb
How do I hide the three topmost boxes labelled "Genre" "artist" "Album"?
I only want to see the Playlist.
That's called the Column Browser. You need to disable it.
Edit > Preferences > Layout (1) > main panel > show column browser - remove the checkmark.
How do I save/export the current layout (and import it into another MusicBee installation on my notebook)?
I'm not 100% certain how to go about that. You could try copying the MusicBeeSetting.ini file and then copying it to the new installation as a test. The file is found in your AppData folder under MB's install directory (if you have the portable version.) Or for the non-portable version, C:\Users\<username>\AppData\Roaming\MusicBee\StartupSettings.ini.
You can experiment with this and see if it works, or wait until another forum member corrects me. If you're going to experiment, save or copy the original MusicBeeSettings.ini under a different name so you can easily go back to it if my method doesn't work.