I No, those forms are not correct in French; you have to use nouns, and at that there is no way to use the adjective "aucun" in this case; since the person is giving an order without using a verb, "pas de" is the usual negative locution; this is one of the usual formats of indications given by means of signs.

• Pas de rots, de renvois, ni de stockage de nourriture dans la bouche.

Otherwise, the use of "aucun" becomes possible.

• Je ne veux entendre aucun rot ou renvoi, et je ne veux aucun stockage de nourriture dans la bouche.

Note: 1/ "stocker de la nourriture dans les/ses joues" is not idiomatic because of the definition.

2/ The use of both "rot" and "renvoi" would be strange in French, because those words are synonyms. The same applies to English, as far as I can see; I don’t quite get the idea in this repetition.

(TLFi) Pop., fam. Émission bruyante de gaz stomacaux par la bouche. Synon. éructation, renvoi.

(TLFi) A. − [Chez l’homme] 1 Partie latérale de la face délimitée par le dessous de l’œil, la tempe et le menton.

II No again, the same principles apply.

• Pas de démarchage, pas de colportage

1 – Regarding the vocabulary:

• Aucun being a determiner it can only be used in front of a noun, it cannot be used in front of a verb, and although the French infinitive can in some cases be used a a noun, it is not the case here. To negate a verb in French you have to use ne pas.

• I would translate "belch" by éructer (the educated word) and "burp" (marked as slang in the OED) by roter (more familiar).

2 – There are numerous ways one could say what you want using an infinitive in French, with nuances though, and I expect none as neutral as the "no+gerund" form in English.

• a) Il est interdit d’éructer, de roter et de trop se remplir la bouche de nourriture¹.

• b) On est prié de ne pas éructer, de ne pas roter et de ne pas trop se remplir la bouche de nourriture. (Or : prière de…)
  Which is slightly more polite.

• c) Ne pas éructer, ne pas roter et ne pas trop se remplir la bouche de nourriture.
  Would sound neutral and closer to the English "no+gerund" form and would be grammatically correct but I expect it will rarely be found. Personally I would expect something after that, such as sous peine de punition².

3 – A sign with "No soliciting" would be Démarchage interdit. Il est interdit de démarcher is correct French but I expect it would hardly be found.

I am not sure I understand what you mean when you say "as a question". In English "No soliciting?" is a question in as much as you end the sentence with a question mark. The same can be done in French, just replace the period by a question mark.

Edit after the question has been edited.

Now that the question has been modified not to ask about
On ne rote pas, on n’éructe pas, on ne stocke pas de nourriture dans ses joues! (affirmative) but about On ne rote pas, on n’éructe pas, on ne stocke pas de nourriture dans ses joues? (interrogative) I will add:

1- What I stated and explained in my original answer still holds and you cannot use aucun, which is a determiner, in front of a verb. If you want to write a question you can, just as you do in your English sentence, add a question mark and when speaking modify your intonation so that it follows the question pattern.

Ne pas éructer, ne pas roter et ne pas trop se remplir la bouche de nourriture ?

2- In a comment you say that you wanted to ask "Could you [do something] no belching, burping, or excessive chipmunking?", which is yet something different. But the vocabulary and grammar remarks still hold.

• If your infinitive were directly dependent of pouvoir you would have :
  Pourriez-vous (pourrais-tu) ne pas éructer, ne pas roter et ne pas trop te (vous) remplir la bouche de nourriture ?

• But the infinitives éructer, roter,… depend on an action (modified by pouvoir) and what you really mean is "can you do something without belching…". "Without" in French is sans and you cannot omit sans.

  • Peux-tu manger en silence, sans parler ?
  • Pourrais-tu (pourriez-vous) [faire ceci] sans éructer, sans roter et sans trop te (vous) remplir la bouche ?

3.
In the same way you could say :

• Sans démarcher ? Sans colporter ?
  Peut-on distribuer des publicités sans démarcher, sans colporter ?

^{1The craze of food contests has not yet reached France and "chipmunking" has no standard equivalent. If I wanted to stay close to the image of the chipmunk I would say se gonfler les joues de nourriture but I am not sure a French native who hasn’t heard of food contests would know what I am talking about.}

^{2 One could read on a street sign :
Jeter son masque sur la voie publique est puni de 68 € d’amende.}

Without more context it’s hard to tell. If by a question you mean something like:

"Could you drink a gallon of beer, no belching?"

You’d use in French this form:
"Pourrais-tu boire sans roter, sans baver, etc?"