Actually, no. But @Banno has explained why not better than I could. You have to fill in something that could actually be true or false, in order for it be a proposition.
Strictly, not. The variable ‘x’ is unbound. It needs a quantifier or an instantiation.
So U(x)(x=x) is a proposition. And so is 5=5.
Ha! Jay answered while I was typing.
What he said. 
Instantiation, right, something it’s about.
It just depends how it is being used. @Banno has himself been using it as if it were a tautological proposition:
In other words, @Jay is quibbling.
Explain this to me please
Apparently we are taking “x=x” to signify something like the law of identity. @Banno wants to read that claim of identity as stipulated, and this is in line with his formalistic tendencies; @TheEudemian is asking about the (metaphysical?) basis for such a judgment. So there is a difference over whether x=x requires justification, presumably because there is a difference over whether x=x represents something beyond mere stipulation.
More simply, the phrase “x=x” is being used in slightly different ways, or at least with different presuppositions. What Banno is doing is characteristic of the Wittgenstenian approach, where one claims that certain affirmations require no justification.
Why was it stipulated? How is it being used?
I’ve read it. It isn’t clear.
I agree that it doesn’t make any sense. But for @Banno the symbology of logical formalisms does not need to answer to natural language. He cares a great deal about logical formalization using logical symbols. “Formalization of what?,” you might ask? That’s a great question, because if the logical formalisms are themselves stipulated for no further reason, then its not clear what it is that is being formalized. Presumably x=x is a formal way of speaking about a truth of identity that actually holds in reality.* If it is divorced from reality then it’s not clear what use it is.
On my view if one cannot say, in plain language, what their formalism is getting at, then the formalism has become an impediment to philosophy. In philosophy “it’s just stipulated” is not a proper response.
* Note too that the one who relies entirely on formalisms will have formalized definitions of ‘truth’, ‘proposition’, etc. Their entire language is internal to their system.
Banno knows logic (more than anyone else in the world) 
Stipulated suggests I could stipulate \neg x = x.
Sure, and then we would have to ask whether one stipulation is better than the other. That would get back to @TheEudemian’s questions about justifying one’s affirmations, namely the affirmation that “This should be stipulated” (which is implicit in the person who does the stipulating).
So my question stands, clarification is needed why x = x.
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“=” is a symbol used in some logics to say that the items on either side of the “=” can be substituted, one for the other, without changing the truth value of a sentence.
Those logics are “extensional” logics.
Here’s an example, from mathematics: 5+5=10, and 7+3=10. We also have 10=10; substitute “5+5” for the “10"on the left side, and “7+3” for the"10” on the right, we get 5+5=7+3.
That has nothing to do with Wittgenstein. It’s just standard logic.
There is an addition, called Leibniz’s Law, which says ∀x ∀y (x = y → (φ(x) ↔ φ(y); that is roughly parsed, if two things are equal, then any statement about one is true of the other. This is a seperate issue. `
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This would make sense if x = y. I can substitute x with y. Saying x = x doesn’t afford us that facility. Why would something equals itself be important to logic?
Not sure what you are asking.
x=x set out part of what “=” does. it’s reflexive; it applies to itself. It’s not all that informative.
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Yes, you could. That would change the way your are using “=”. It would no longer be what we call identity. It wouldn’t be a question about which is better, only about which is being used.
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x=x because that’s what “=” does.
In first order logic it can be introduced b y taking it as an axiom, in axiomatic systems. Or it can simple be stipulated, in a natural deduction or sequent system.
Folk whose only experience is with axiomatic presentations sometimes mistake the axiom for an empirical claim. It isn’t. It is part of the specification of the identity symbol. It doesn’t need any further justification.
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A stipulation is a proposition that is assumed as true without justification: every belief that you build from that stipulation would thereby be unjustified. You are accepting the ‘dogma’ option I gave in the OP, which is just to bite the bullet.
What you claimed, from what I understand, is that the fact that every belief ultimately is unjustified merely exposes the lack of absolute certainty we can have about our beliefs (as opposed to nullifying them). If that is what you are saying, then you are not contending with the Trilemma.
There’s a difference between a belief ultimately being justified with a belief that is unjustified vs. taking a belief as true with certain pragmatic stipulations. We may assume that ‘x=x’ when disputing calculus for the sake of brevity; but that’s different than grounding the justification for a calculus proof in ‘x=x’ as a matter of truth that purely assumed. This is exactly why @Scorpion is absolutely justified in parodying your argument with this:
If you can justify your beliefs ultimately with baseless assumptions, then they can too; and neither would have any rational recourse to resolve it.