Either or neither

In The Merchant of Venice Lorenzo says:

The man that hath no music in himself,
Nor is not mov'd with concord of sweet sounds,
Is fit for treasons, stratagems, and spoils;

In the second line Shakespeare has chosen to break one of those rules that often incites the grammar dogs to sic. He starts the line with "Nor" when there has been no "neither" to set up the form. The "neither...nor..." construction may be more strictly enforced than "either...or..." because of questions using "or" that can't use "either" without sounding grossly awkward or being ungrammatical.

"Are you going to pay with check or with cash?"
"Can you handle that or should I help you lift it?"

When "or" is used without "either" as in

Shut up or I'll give you something to cry about.

some might argue that the "either" is understood and left out. Much as the [you] is left out of imperatives like "[x] Close the window." In that case we would have a null [either] in

[x] Shut up or I'll give you something to cry about.

I've never looked at the syntax of these statements so I'm just guessing here. "Either" would be perfectly grammatical in the sentence which supports a shared structure.

Where would we put the [x] in the questions? Let's move the copula back into it's declarative position in the check/cash question and use the "either".

You are going to pay either with check or with cash.

When I move the copula over for the question the either sounds questionable to me.

?Are you are going to pay either with check or with cash

What keeps it from being completely ungrammatical is the possibility that it becomes a yes/no question conditional on only one of the payment methods being true. Ie paying with check gets a yes and paying with cash also gets a yes.

But we reanalyze the sentence, remembering to account for every underlying representation (UR) missing from the surface representation (SR). Our underlying form would then be

[Either [you are going to pay with check] or [you are going to pay with cash]]

The declarative form of "can you handle that or should I help you?" helps us to see this UR.

[Either [you can handle that] or [I should help you]]

We see then that while "or" always stays in place "either" usually doesn't. The following still sounds awkward to me but not ungrammatical.

Either are you going to pay with check or are you going to pay with cash?

This emphatic use of either sounds like a demand for a decision. Something along the lines of "either come inside or close the door" when a lollygagging visitor lets the elements into the house.

The "either" sounds almost ungrammatical in the following:

??Either can you handle that or should I help you?

Apparently either doesn't really like sitting next to a raised copula. But "neither" almost always does. Even in a declarative the "neither" seems to pull the copula forward.

You are neither smart nor funny. cf Neither are you smart nor (are you) funny.

We note here that simply raising the copula does not create a yes/no question. We also have to put the "neither" after the subject to form the question.

Are you neither smart nor funny?

And at this point I realize that the post is getting long, and I'm getting farther away from an explanatory analysis. I would keep writing, but I started this post thinking about pursuing double negatives. Now that I've let that fish off the hook I'm going to reel in and recast later. Any syntacticians who can contribute to some actual understanding are welcome to point out my mistakes and oversights.