A short(ish) puzzle about have (got)-possession in English

After today's discussion of possession, I wanted to write down a short (I hope) puzzle that I have been thinking about in connection with have-possession in English. To keep it short, I'll mostly avoid fleshing out all the implications and just treat it as a series of puzzles, which I think are relevant to a lot of what we have talked about. (Now that I finished, I see that it would probably exceed the word limit for Snippets...) 

The starting observation is that (1) and (2) are basically equivalent. 

(1) She has a lot of money.
(2) She has got a lot of money. 

In my (American) English, got is very different from gotten, so (2) cannot be eventive/inchoative. It is strictly possessive. In (1), have normally cannot undergo subject-aux inversion (nursery rhymes aside); see (3). But in (2), it can; see (4). 

(3) *Has she a lot of money? / Does she have a lot of money. 
(4) Has she got a lot of money? 

So let's call this puzzle 1: (2) has auxiliary have along with a special got which is present tense possession—not semantically perfect in any sense. 

Now, puzzle 2 is that (2) cannot in any way be past tense for me. So (5) is basically out for me:

(5) *She had got a lot of money. ≠ 'She had a lot of money (when she was a kid).' 
(6) *Had she got a lot of money. ≠ 'Did she have a lot of money (when she was a kid)?' 

Whatever you think of (5) and (6), for everyone I have to talked to, they cannot mean what the translations after the ≠ say. So puzzle 2 is: the odd have got construction cannot be in the past tense. Actually, it's broader: it must be in the present tense. 

(7) I believe her to have got a lot of money ≠ I believe her to have a lot of money.
(8) She will have got a lot of money. ≠ She will have a lot of money.

And so on. Finally, one more puzzle, which we have talked about in the YGDP (https://ygdp.yale.edu/phenomena/bare-got): many speakers allow do-support with this possessive got. But when they do, the same restrictions on tense hold:

(9) Does she got a lot of money? = Does she have a lot of money? 
(10) *Did she got a lot of money? ≠ Did she have a lot of money? 

Once again, it requires this present tense: 

(11) *I believe her to got a lot of money.
(12) *She will got a lot of money. 

One way to put puzzle 3: possessive got requires present tense and either a dummy T/do, or a dummy have. The "T/do" is because when you don't need do-support, you can often get no auxiliary, but it has to be present tense:

(13) I got a lot of money. 

But there is at least one more aspect to the puzzle, which is that for me, I must have an auxiliary in the 3rd person singular:

(14) She does got a lot of money. (stressed do)
(15) She's got a lot of money. 
(16) She has got a lot of money. 

So let's say that for me, I can delete or fail to pronounce 've in a way that is part of a distinct pattern (maybe). So the puzzle is puzzle 3 (revised): possessive got requires present tense and a dummy auxiliary, which can be do (if independently licensed), or a dummy have. 

So even for speakers who reject the do at the heart of puzzle 3, puzzles 1 and 2 are, I think, very real. And one other thing that makes it even more puzzling: everything I said above applies to the modal use of have to versus got to / gotta, in some cases even more sharply (but not necessarily everything we have wanted to call main verb have). 

So my very vague question in connection with the conversation we had about the (possibly unpronounced) "root" of a word like have is this: Is there some way of understanding the pattern above that connects the low position of got and get more generally (e.g. in get-passives) to the fact that main verb have looks like auxiliary have even when the former is linearized "low" and the latter is linearized "high"?