Firstly, note that the best gift [that] I [have] ever have isn't a "sentence" - it's a "noun phrase" (within which the "head noun" is gift). Also note that it's perfect okay to include or omit either/both the "relativiser" that and the "auxiliary verb" have, and this has no effect on the meaning (it's entirely a stylistic choice).
BUT the second instance of have isn't an auxiliary. It's a normal verb usage, with the meaning get, obtain, acquire, receive,... So our choice is between...
1: This is the best gift I ever get
2: This is the best gift I ever got
3: This is the best gift I will ever get
In all those examples, the optional word ever provides emphasis (drawing attention to the fact that I never get or got or will get any gift better than this on any occasions whatsoever).
In most contexts, Past Tense (best I ever had) is the correct choice (in which case you're not necessarily ruling out the possibility of getting an even better gift in the future). You're saying that at no time in the past did you ever have anything better.
But note that Present Tense (best I ever have) doesn't exactly refer to present = now = time of speaking. It's the "timeless" use of the verb form, implying "never at all" (in the Past, Present, OR Future). And that's probably not the intended meaning.
It's possible to be even more emphatic by using could instead of will for the "future" reference in #3 above...
4: This is the best gift I could ever get
(there never was and never could be a better gift)
...and it's possible, though a bit clunky, to use that form for the "Past up until Present" context...
5: This is the best gift I could ever have got
(no past or current gift is better, but feasibly there might be a better future gift)
Why does that construction exist?
Because in spoken language, should have and should of are indistinguishable (when enunciated as contracted should've). And some native speakers are so ignorant they don't understand that basic syntax requires an auxiliary verb after should, not a preposition.
But note that some perfectly competent Anglophones may deliberately write the incorrect version sometimes as a facetious usage. OP's cited Facebook poster probably isn't very competent, given that "next time" is future, so the auxiliary have (part of a "Present Perfect = Past" verb form) shouldn't be there anyway. But even people who know perfectly well that it's "incorrect" sometimes write things like a whole 'nother ballgame for much the same reason.
I don't know why OP wrote it's not correct to use should have. That's exactly what the Facebook writer should of1 written!
1 In case it's not obvious, that's me being facetious! But actually whereas one should've = should have done something in the past, when talking about a future "next time" it's what one should do (infinitive do, not have + Past Participle done).
Best Answer
It's wrong to say
You should say
Here, "what it does" is not a question, it's a Noun Clause.
"What it does?" is not a proper question on its own. We don't see, for example, subject-auxiliary verb inversion in it.