Quoi is only used as "pseudo second negator" with the verb savoir. I write "pseudo" because there is actually kind of an implicit pas in this construction, unlike the negations with guère, jamais, nullement, etc where there can’t be a pas.

None of your attemps with a different verb would work.

Quoi can be used in negative statements but never as a negator:

— Je mange pas de [inaudible].
— Tu manges pas quoi ?

Here quoi is a pronoun replacing something that wasn’t heard or understood.

Note that your question was legitimate. While a native French "knows" what can be a co-negator and what can’t, there is no obvious difference at first sight between je ne vois personne and je ne sais quoi.

In early French, the first occurences of je ne vois personne were technically built like "I don’t see [a] person" and je ne sais quoi. That’s only after time that personne split into two words belonging to different categories, one still a positive substantive (Une personne, someone) and another generally negative, the pronoun personne (nobody) but sometimes still positive (anyone).

Such a split never happened with quoi, so it stays a "positive" pronoun even in je ne sais quoi (I don’t know what) and accepts a true co-negator like pas or plus:

Je ne sais pas quoi.

Je ne sais plus quoi.

With personne, pas is excluded (or at best, reverses the meaning) :

*Je ne vois pas personne. (*I don’t see nobody)

but plus is fine:

Je ne vois plus personne. (I don’t see anyone anymore)

Quoi is not part of the negation, it doesn’t function with ne, it can’t be grammatically nor semantically compared to pas, point, nullement, guère, etc.

In the sentence je ne sais quoi dire the negation is ne…pas where pas is omitted. The answers to this question might help you go further with the omission of pas.

Quoi introduces a dependent clause, it is a pronoun, it is the direct object of the verb dire.

The use of quoi depends on what follows in the sentence.

• Je ne sais (pas) quoi dire.
• Je sais quoi dire.

If dire had a subject then you could not use quoi, you would use ce que:

• Je ne sais (pas) ce qu’il dit.
• Je sais ce qu’il a dit.

You could also have :

• Je ne sais (pas) où aller.
• Je ne sais (pas) comment dire ça.

Although savoir is not the only verb with which pas can be omitted in a negative sentence, I think it is the only one in the list which admits quoi (or ce que) to introduce the following dependent clause.

This page has a list of the verbs that can omit pas.