I'll share my opinions on the subject.
Imagine the below example as a "diagram" of what the different planes are theoretically supposed to be.
Astral Plane
Etheric/Ethereal Plane
Physical Plane (Lucid Dream)
As you can see, a Lucid dream is a physical phenomena. What you are experiencing are physical hallucinations (like any other dream). The difference, of course, is that you are aware the things you are seeing/touching/hearing etc are just creations of one own mind.
An etheric/ethereal projection is supposed to the consciousness literally leaving your body and being able to move around in a place that is very true to physical plane (some even think it is the physical plane). So in a way, the Etheric/Etheral Plane is a buffer between the physical plane and what is known as the astral plane.
An astral projection refers to when the mind leaves the body, but goes to a different "dimension" or plane of existence that is removed from the physical plane all together. What you are seeing in one of these projections is supposedly real, just not in regard to the physical plane of existence.
Given these rough breakdowns, and Both an EP and AP would qualify as an OBE (considering that your consciousness is indeed leaving the body), but an LD would not.
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Here is my interpretation of the planes, and how they relate to one another:
To contrast the first example, this graph shows EPs and APs as both being of high detail, however with EPs being accurate to the physical plane and APs not. And lucid dreams can be found somewhere in between, but as you can see this is a continuous scale of realism/accuracy and not "levels" or defined "stages" of different phenomena. If you frame the concepts in this way, EPs and APs can all be considered types of LDs.
As far as addressing question 2, it is my personal belief that all three of these phenomena are creations of the mind. However, I am open to the concept of ESP in general, and so I suppose a theoretical explanation of an EP could be a psychic lucid dream. But most of this is just conjecture really. I have very little research to back up these claims at this point. But I am working on it.
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