$\{f \in C \mid f(x)>d$ for each $x\in D$ for some $d\in \mathbb R\}$
Do you read the above set as "the set of functions $f$ in $C$ such that there exists $d$ such that for each $x, f(x)>d$", or do you read the $d$ as depending on $x$ i.e. "the set of functions $f$ in $C$ such that for each $x$, there exists $d$ such that $f(x)>d$" ?
I always read the quantifiers in reverse order when they are written behind the proposition (like in the set above), but I've been wondering if this is universal or just me.
I don't think there is a universally accepted meaning for things that look like formulas but with the quantifiers in the wrong place (that is, not in front of the part they apply to.) I think you should usually avoid such things.
In natural language I think it is fine to put a quantifier at the end if there is only one quantifier, as in "let $E$ be the set of natural numbers $n$ such that $n = 2m$ for some natural number $m$." Even with more quantifiers there may be some cases where it is okay to put them in at the end or in the middle but I think the best way to avoid ambiguity is usually to put them at the beginning.