Call-by-push-value's denotational semantics of "thunk diverge"
Yes, your reading is correct.

$U$ forgets that something is a domain, but does not change the underlying carrier set or the poset structure. The least element is still there.

It is also the case that $U \circ F$ can be equipped with the structure of a monad, as you say.

And yes, predomains need not have a least element.