Question number 1: Is this correct?
Question number 2: If yes, would be correct if for the conclusion I use x instead of y? For the context, it seems clear that the right choice was y. However, as x and y are variables I was wondering if it would be the same. Thank you so much for your help!
Translate to logic symbols:
There is a man whom all men despise. Therefore, there is a man who despises himself.
Premise:
$\exists y \forall x (Mx \land My \land Dxy)$
Conclusion:
$\exists y (My \land Dyy)$
On
[EDIT: the question changed after I posted this answer. the original premise was $(\exists y)(\forall x)(Dxy)$.]
If you've defined $Dxy$ as "$x$ despises $y$", then your premise and conclusion are correctly translated into symbols.
You're right that $(\exists x)(Dxx)$ is equivalent to $(\exists y)(Dyy)$. That's because there is a rule of the predicate calculus which states that we can rename the "bound variable" $a$ to $b$ in $(\exists a) \phi$ without restriction, as long as $\phi$ doesn't contain any free [unbound] occurrence of the symbol $b$. [A "bound variable" is one which appears inside the bracket of an $\exists$ or $\forall$.]
That is to say, "the name of a variable doesn't matter".
Aside: We do need $b$ not to appear free in $\phi$. Otherwise, from the line $(\exists a)(a < b-1)$ we may relabel $a$ to $b$ and deduce $(\exists b)(b < b-1)$. But this latter formula is obviously not true even though $(\exists a)(a < b-1)$ may be true if $b$ is chosen correctly.
The premise is not correct. It should be $\exists y (My \land \forall x (Mx \rightarrow Dxy))$