Concerning problem 2, there are two ways to formulate "no one has property P" using quantifiers: (1) for every x, it is not the case that P(x) and (2) it is not the case that there exists an x such that P(x). The property P here is that x visited every state. This can be formulated as "for every y, x visited y".
Problem 1 is easy if there were no exceptions (Oregon and Nebraska). Then one would say, "There exists a person x such that for every state y, x visited y". To make sure that the state y is neither Oregon nor Nebraska, we could say, "for every state y, if y is not Oregon and y is not Nebraska, then ...".
For problem 3, I would expect the instructor to give the basic relations that can be used in translation. Without this information, if I wanted to play smart alec, I would call the translation of the whole proposition by one letter P.
In problem 3, it makes sense to say V(x, y) again denotes "x visited y", but y is now a person. Also, it makes sense to consider a set of places with NYC and possibly USC being constants from this set. Finally, one needs another relation L(x, y) for "x is in y" where x is a person and y is a place. Alternatively, one can consider a ternary relation V(x, y, z) to mean "x visited y who is in z".