Caroll's paradox => Rice theorem?
Well, any true statement implies every other true statement, so in that vacuous sense, I suppose one implies the other.

But no, I wouldn't say that Carroll's paradox implies Rice's theorem in any interesting or meaningful way.  Carroll's Paradox is an explanation of why we need modus ponens as an inference rule in propositional logic and can't replace it with a finite list of axioms.  It's nothing fancier than that.  As such, it refers to what is true in a logical theory.  It refers to what inference rules or axioms we need.

Rice's theorem is different.  It talks about what is decidable, not what is true.   It talks about what can be computed on a Turing machine, not about the inference rules or axioms we need.

Rice's theorem doesn't say "it's not possible to formalise the demonstration of a non-trivial property of a recursively enumerable language within the same given language".  It has nothing to do with "within the same given language" (it's not even clear what that would mean; a language is a subset of $\{0,1\}^*$ and doesn't demonstrate anything), but rather with that can be computed with a Turing machine.  Rice's theorem talks about what a Turing machine can decide.

Rice's theorem doesn't say "you will necessarily need a metalanguage to define modus ponens".  (I don't even know what a metalanguage would mean in this context.)  It has nothing to do with what you can define.  Rather, it has to do with what a Turing machine can decide.

In general, I warn you against trying to reason about mathematical statements using English language and analogies.  That is notoriously error-prone.  Rice's theorem is a mathematical statement with a precise mathematical meaning.  At a suitable level of imprecision and vagueness, I suspect everything can seem similar and connected -- but when you look at the specifics, the situation often becomes richer and more interesting than that.