Floating quantifiers in X-bar theory: "the men all have gone"

Question

In chapter 9 of Syntax: A generative Introduction (2nd ed), Carnie shows that we can solve some problems by generating subjects in Spec of VP and letting them move to Spec of TP. In the first challenge problem set, he shows that this also allows for a nice analysis of floating quantifiers, by considering the two examples:

1. All the men have gone.
2. The men have all gone.

The D-structure tree for this sentence is:

The claim is that either DP could be moved to Spec of TP, yielding sentence 1 and 2 depending on which is moved.

But what about:

3. The men all have gone.

^{(I am not a native speaker and this seems kind of odd, but "the men all have" gives plenty of exact hits on Google. According to this—not sure how reliable—this construction is allowed with the verb to be.)}

It seems that here, "all" has become an adverbial adjunct of TP:

Is that correct? Which rule allows this movement?

A secondary question I have is that with this apparently over-simplistic presentation in the assignment, "all" should not really be called a floating quantifier: it does not float but stays in the same position. Only with my analysis of (3) is there any movement of the floating quantifier. And, of course, on the surface level "all" seems to float if you just compare (1) and (2) (and (3)). So where does the term "floating quantifier" come from, and is it really accurate?

Answer 1

'The men all have a {noun}' is fine, but 'the men all have {verb}ed' is not.

The rule is probably the same one as the one that has us say 'they/we have all {verbed}' rather than 'they/we all have {verbed}'. Also, the other way around for a noun, 'they/we all have a {noun}' rather than e.g. 'they/we have all a hat'.