This is an initial attempt at answering, and the answer may evolve.
- The verse itself is a synthesis of romantic and transcendental love
Joe Campbell would likely have talked about the Amor/Roma connection, which might also be expressed as Eros/Agape.
Could have many referents. I thought about Arthur/Guinevere, and in that context is could be a reference to betrayal. (Arthurian legends have many Christian elements, although Tristan&Isolde is a more direct expression of Amor/Agape.) But it seems more likely this is again a reference to the Old Testament, probably a metaphor.
- "She cut your hair" is definitely a reference to Samson & Delilah
which is an allusion to the bible, consistent with the song title. In this context, it probably reflects the willingness of the pursuer to surrender their power to the object of desire, regardless of consequence.
- "You saw her bathing on the roof" is obviously a reference to David and Bathsheba, which is about lust, sin and redemption
[Thanks to Mithrandir for reminding me of this!]
- And from your lips she drew the Hallelujah
as the final line would seem to support this hypothesis that the agony and ecstasy of love, in this case, are welcomed and redemptive.
- Your faith was strong but you needed proof
Is a restatement of a very old philosophical question in Christianity. Here I think it could refer to his faith in love, on a transcendental level, but the need for confirmation on a physical level.
Note: "You saw her bathing on the roof/Her beauty and the moonlight overthrew" made me think of Diana/Artemis, but this may be per my background in Classical mythology.
Bathed may have a literal meaning, but for me it is clearly an allusion to the moon goddess, for whom being spied bathing is a central myth. Acteaon is torn apart by his own hounds, which can be a metaphor for the agony of desire, or the inability to contextualize such transcendental beauty. The story also has an element of human sacrifice to appease the goddess. Clearly, Cohen's object of desire is a goddess to him, worth any sacrifice or torment.
I have no doubt the highly literate Cohen was aware of this story, and I think it's also extremely probably he was familiar with The White Goddess, which among other ideas, casts poetry as sort of a sacred language. The matriarchal elements are consistent with Cohen's surrender, romantically and spiritually, to the object of desire.