Do there exist any circumstances where any verb other than read is used with between the lines? That is, is between the lines an independent and complete idiom, or is it incomplete and meaningless without read?
The phrase "read between the lines" is a set phrase, with a literal and metaphoric connotation. You can use it to refer to an assessment of a work in print (literal) or use it to mean the act of examining the subtext of a strategy, a speech, etc.
You very well could say, "When we attended the talk, we were listening between the lines," but this would be a nonce coinage, playing off the standard version with "read." Same with "look," "feel," etc.
It can be used with other verbs with a meaning related to "read", but "read between the lines" is the most common. Even in extended uses there is reading or writing (or at least interpretation of some kind) happening somewhere. For example:
She read that section again. Was he trying to say something between the lines? Hinting about Sam?
from the short story Of One Mind by Shane Tourtellotte.
This is from an article The Future of Medieval Church History by John Van Engen (Church History 71(3):492-522, September 2002):
Still, between the lines it discloses a world filled with spiritual forces and human paradoxes
Here "it" is a story from the Dialogues of Gregory the Great, so the phrase is still being used in the context of reading, although not with the verb "read".
If used as an idiom, then the full phrase is "read between the lines". Of course the phrase "between the lines" could always be used literally as opposed to figuratively, but it wouldn't have the same meaning.
It's such a well-known idiom that shortening it to "between the lines" would generally be understood, even without any specific reference to reading.