Existence of particular/specialized halting oracles
Yes. Given a machine $M$ and input $I$, one of the two things happens:

$M(I)$ halts.
$M(I)$ does not halt.

We may not be able to predict or compute which of these happens, but one and only one happens simply by rules of classical logic.
Now to show that $H_{M,I}$ exists we consider two cases:

If $M(I)$ halts then $H_{M,I}$ exists because we can take it to be a machine that immediately halts and outputs &quot;yes&quot;.
If $M(I)$ does not halt then $H_{M,I}$ exists because we can take it to be a machine that immediately halts and outputs &quot;no&quot;.

Note: we proved existence of $H_{M,I}$, but we did not give a way of computing $H_{M,I}$ from $M$ and $I$, which is impossible as that would give us the halting oracle (exercise).
So, what Yuval said.