Two questions on MSO
You are asking two questions. The first question is about logic more generally. You are wondering whether a statement of the form
$$ \exists x \, \mathrm{True} $$
is valid, although $\mathrm{True}$ doesn't mention $x$. There is absolutely no requirement in the syntax of logical formulas that  you use all the variables you quantify over. This similar to functions which don't depend on all of their arguments. Do you object to the function $f(x) = 1$?

As for your second question, monadic refers to the fact that you are only allowed to quantify over unary predicates. The language itself can freely contain $k$-ary predicates for any $k$. Compare this to first-order logic, in which you are not allowed to quantify over any predicates. Would that mean that first-order logic only allows for 0-ary predicates?