I know "something don't exist" and "something exists" are both correct, but when comes to spoken language, sometimes we say "not exist(s)" for short..
A: "does something exists?"
B: "not exists."
So which one is right?
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Neither is a correctly formed utterance. If you're asked:
Does photographic memory exist?
The shortest answer is simply "No."
If you want to use a complete sentence, replace the subject with the pronoun it and add the dummy auxiliary do so you can negate it and omit the verb:
No, it doesn't.
I'm afraid that neither of the options you've given is correct.
"<something> don't exist" is not Standard English. As TRIG pointed out, this should be "<something> doesn't exist", because <something> is singular.
"something exists" is correct.
"Ain't no such thing" is common in spoken English, but "Ain't" is not in Standard English. (Also, this use of a double negative is incorrect per Standard English.)
"That exists" and "That does not exist" are Standard English, if the implied subject is singular. The plural forms are "Those exist" and "Those do not exist".