For LiveView contrast AF the system is just racking the focus in order to maximise the difference between pixels (though it may be aided by extra logic). For phase AF the image is split into two, and the phase sensor can detect by how much, and in what direction, the two images are out of phase. Hence AF can be much faster. As you say, little/no detail in the image (possibly due to everything being way out of focus) means there's not much detail to help the phase detection work - hence the apparent hunting.
Yes, looks like we agree on this.
As for the live view AF, mine is different, at least on my 70D, it is also phase detect AF.