A couple of things...

The first one is pretty simple, just tick "direct search" and notice that "advanced query syntax" is still enabled. If you tick "advanced query syntax", it disables "direct search", but not vice versa. I'm not fond of disabling the boxes because then you have to untick one in order to tick the other, but whatever it does, the behaviour should be consistent at least.

The second is probably several things:
Tick "advanced query syntax" and search for something.
Notice that on the results page, direct search is still enabled when it shouldn't be.
You can then tick direct search (making both enabled).
Untick "advanced query syntax" and it will also untick "direct search".

Maybe we should just have radio buttons or a dropdown... it'd mean including the default indexed search as an option, but at least it would be clear that the options are mutually exclusive, and it wouldn't require any js either...