You seem to be mixing up two approaches: you're using both existential quantification and also material implication, when you only need one.

Is this homework? You should try to figure it out on your own (and also not be posting the questions online).

If not, if you just want to talk about the ideas behind predicate logic, we can discuss it here, but be aware this will NOT necessarily result in the "right" answers for your class which will use one specific approach that won't be the same as other classes or what people on the internet say.

"→" is material implication. It's similar to English "if", but in a very specific sense. It's a relationship of truth conditions.
https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)

The sentence you're thinking about is complicated. It can be reduced to a simpler structure by substitution (eliminating some of the negation, for example), but I'd suggest starting with a basic sentence and working it out from there.

Notice also the relationship between material implication and negation mentioned at the start of the Wikipedia article:

In propositional logic, material implication[1][2] is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that either form can replace the other in logical proofs.

To begin, I'd suggest trying a simpler sentence like "Nobody sleeps", and working up from there.

That's one way to represent it (though you could replace 'for all X' with symbols), among several options, and assuming the most natural scope reading for the original sentence.

There's actually a built-in addition to the forum for these symbols (at least a few: I could try to add more by request if needed):
http://linguistforum.com/phonetics-and-phonology/type-ipa-within-the-forum!-(instructions)/
(You'll need to be in the 'advanced' editor, rather than just the quick reply mode below a post.)

Alternatively there aren't many other options than just copying and pasting. I usually go to this page if I need to find something unusual:
https://en.wikipedia.org/wiki/List_of_logic_symbols