Formalization of proofs and computational complexity paradox?
$\def\mc{M_\mathit{const}}\def\mp{M_\mathit{paradox}}$Let me for the record write up the answer to Q1, so that it doesn’t live only in the comments.

The reasoning given in steps 1–5 in the question is correct in the real world. Thus, $\mc(\mp)$ outputs NO, and $\mp(x)$ halts in constant time, but there is no short enough proof of this in ZFC.

When trying to formalize this argument in ZFC, the problematic step is 2: here, we need to assert, in ZFC itself, the implication 


  (1) If ZFC proves ‘$\mp$ halts in constant time’ by a proof of length $\le|\mp|^2$, then $\mp$ halts in constant time.


In general, statements of the form


  (2) If ZFC proves $\phi$, then $\phi$.


are known as instances of the reflection principle for ZFC, and by Löb’s theorem (a generalization of Gödel’s 2nd incompleteness theorem) they are never provable in ZFC itself, except for the trivial case when ZFC already proves $\phi$. Notice that the consistency statement for ZFC is a special case of the reflection principle, for $\phi=\bot$; in general, reflection principles are strictly stronger than consistency.

In our case, $\phi$ is the $\Sigma^0_2$ formula “$\mp$ halts in constant time”, so step 2 apparently makes an appeal to the $\Sigma^0_2$-reflection principle for ZFC, unprovable in ZFC.

However, we have to our disposal an extra restriction that we ignored so far, namely that we have a bound $\le|\mp|^2$ on the length of the purported ZFC-proof of $\phi$. Crucially, at this point, $\mp$ is a fixed explicit Turing machine, hence this bound is a standard integer constant (and likewise, $\phi$ is a fixed sentence). Thus, (1) is actually an instance of the finitistic reflection principle


  (3) If $\phi$ has a ZFC-proof of length $\le s$, then $\phi$.


This is actually provable in ZFC: if $\phi$ does, indeed, have a ZFC-proof of length $\le s$, then we can just take it, and derive (3) by ignoring the premise; if not, then for each string $|w|\le s$, we can “by inspection” produce a ZFC-proof that ‘$w$ is not a ZFC-proof of $\phi$’, and concatenating all these together, we obtain a proof of (3) (specifically, a proof of negation of its premise).

This brute-force proof will have length exponential in $s$, i.e., in our case, $2^{O(|\mp|^2)}$. There are in fact proofs of (3) of length polynomial in $s$ and $|\phi|$: this was proved by Pudlák [1] using efficient partial truth predicates. The improvements in [2] are only stated for consistency statements, but I believe they apply to more general reflection principles as well. Thus, there are ZFC-proofs of (1) of length $O(|\mp|^2)$.

However, this is still not enough to get a contradiction, as it seems any reasonable proof of (3) needs length more than $s$ (though the lower bounds in [1,2] are not quite that strong). Thus, ZFC-proofs of (1) will need length more than $|\mp|^2$. This cannot be fixed by raising the bounds in $\mc$ or by padding $\mp$, as these will just raise the length parameter $s$ in the same way. Thus, there is no paradox.

So, to summarize: ZFC proves that $\mc(\mp)$ outputs NO and that $\mp(x)$ halts in constant time, but the proofs are a little too long.

It is also worth mentioning that nothing in these arguments depends on ZFC specifically. The same reasoning will work for many other theories, for example Peano arithmetic. (Basically, we need a sequential theory axiomatized by finitely many axioms or nice schemata, see Pudlák’s papers for details.)

[1] Pavel Pudlák, On the length of proofs of finitistic consistency statements in first order theories, in: Logic Colloquium ’84 (Paris, Wilkie, Wilmers, eds.), Studies in Logic and the Foundations of Mathematics vol. 120, 1986, pp. 165–196, doi: 10.1016/S0049-237X(08)70462-2.

[2] Pavel Pudlák, Improved bounds to the length of proofs of finitistic consistency statements, in: Logic and Combinatorics (Simpson, ed.), Contemporary Mathematics vol. 65, American Mathematical Society, 1987, pp. 309–331, doi: 10.1090/conm/065.