The first example is not correct, it should be "or", rather than "nor". "Do not do A or B or C, " for example, in a list.
The second example, though grammatically correct, is not current conversational usage. There is nothing wrong with it, but it's unlikely to be heard in conversation today. Certainly, it's a classical pattern, as in, "Neither a borrower ...
"With" is often not stressed. And especially when followed by a word beginning in "th" it may be shortened.
But the w sound (at least an exhalation of breath) should not be fully dropped, and in writing we wouldn't omit the "with".
It depends on what the speaker actually wants to say.
A. I won't let you through with this.
You can break (A) down like this:
I won't let you [go] through [this metal door] with this (= the bomb/briefcase/belt).
Compare this with
I am sorry sir, but I can't let you enter the building with that gun.
The speaker could also have meant this: