word-request
If we want to say that A depends on none but all depends on A. A needs no one but everyone needs A. What is the best word that fits best this trait?
Let me give my own research that the word 'independent' seems to be covering this particular trait a little less than the other two.
A schtick refers to a comedian's style and choice of topics in general, and could include facial expressions or actions as well as words or style of delivery. It doesn't refer to a specific gag that is repeated often (although a repeated gag might be part of a comedian's schtick). It also often distinguishes a comedian's on-stage role from their actual personality:
Gracie Allen typically played a dingbat character, but that was just her schtick. In real life she was an exceptionally intelligent person.
A running gag is a joke that is repeated in a given work (book, film, tv show, etc.) or series of works. It's not necessarily spoken by the same actor at each occurrence.
"gag routine"(*) is not a commonly used idiom in English.
A signature gag would be immediately understood to have the meaning you suggest, but it isn't an idiomatic phrase, it's just two words used together that mean what you want. However, the phrase is "signature line" is probably more common than "signature gag". For example,
"Take my wife, ... please" is Henny Youngman's signature gag [or signature line].
A catchphrase is another related term, meaning a particular phrase used by a performer (not necessarily a comedian) throughout their career. This could include Henny Youngman's "Take my wife," or John Wayne's calling people "pilgrim". (Thanks @JoeSteele)
I have updated my answer based on several comments I've received.
Although it seems that the word suggester does actually exist as a variant of suggest (per Merriam-Webster), I find it so uncommon that, while understandable, it would give most people some pause on reading it.
I would say that a more common word is proposer.
[Merriam-Webster, propose]
1 : to form or put forward a plan or intention
// man proposes, but God disposesOther Words from propose
proposer noun
In the right context, such as fields to be filled in on a form, the word would be quite appropriate:
Proposal: (____)
Proposer: (____)
Approved: (Yes / No)
Also, I've been informed that proposer is frequently used in formal debates.
As for comparing suggester to proposer, Google Ngram Viewer indicates that proposer is far more common, at least in print:
As I final note, I will add that while proposer might sound strange to some, so too would asker and poster, as used in the question. Just as it would be more common to say the person who asked or the person who posted, it would also be more common to say the person who proposed (or the person who suggested). None of these single nouns are as common as a longer phrase; however, proposer is still relatively much more common than suggester.
Best Answer
All (independent, self-reliant, self-sufficient) are suitable to say: "A depends on none" or "A needs no one".
However, I am not aware of a word to mean: "all depends on A" or "everyone needs A".