Is there a connection between the etheric body and the deep subconscious? Are they perhaps one and the same? If not, what is the relationship between them?

The deep subconscious is your soul essence as a form of consciousness that is in a pure state, although no longer existing as a light being with a full range of ability to manipulate things. It is heavily constrained and attenuated, being in the heavy energy of the physical, held within a physical body that limits its reach in terms of the physical plane. But it can roam about intuitively and will visit the akashic records and inspect what it can see about its own existence in many other timelines, particularly other lifetimes, and gain an understanding of what it has gone through. It realizes there is an important body of knowledge there that will have an influence on the present and the future as well. It worries a good deal about past suffering because that will still resonate, it will be painful to look at even if bad things have happened, and this will stir up fear about what might be coming if the current life seems to be following a similar path that got you into trouble in other times and places. So here again, what matters is your consciousness, not an etheric form of your being, so it would be better to dispense with that term.