The statement
There is a student in this class who has taken every course offered by one of the departments in this school
could be "worded" more clearly. I take it to mean that there is some student $x$ and there is some ("one") department $z$, such that $x$ has taken every course $y$ offered by $z$:
$$\exists x\,\exists z\,\forall y\,(O(y,z) \to T(x,y)). \tag{1}
$$
Notice that in the highlighted English sentence, the "every course" quantifier precedes the department quantifier ("one of the departments"), the opposite of their order in the logical structure.
You didn't symbolize "in this class", so I didn't either in (1), but you should. To keep it simple, let $C(x)$" stand for "$x$ is in this class". A better rendering:
$$\exists x\, \big( C(x) \land \exists z\,\forall y\,(O(y,z) \to T(x,y)) \big). \tag{2}
$$
The "domain" over which variables range is a little unusual for a basic logic example but quite typical of "real-world" situations. It's most naturally thought of as multi-sorted: there are students, departments, and courses, so the actual universe of the interpretation is the disjoint union of these sets. In the intended interpretation of $T$, if $T(x,y)[\mathbf{a}/x, \mathbf{b}/y ]$ is true, then $\mathbf{a}$ is a student and $\mathbf{b}$ is a course; similarly, $O(y,z)[\mathbf{b}/y, \mathbf{c}/z ]$ is true only of courses $\mathbf{b}$ and departments $\mathbf{c}$.
Your statement is very different than what the English sentence (tries to) express. Your rendering:
$$\exists x\,\forall y\,\exists z\,(T(x,y) \land O(y,z))
$$
is equivalent to:
$$\exists x\,\forall y\,T(x,y) \land \forall y\,\exists z\,O(y,z)
$$
which says: there's a student who has taken every course, and every course is offered by some department — two independent statements.
Best Answer
Your answers to (a) and (b) are correct.
Your answer to (c) says: there exist two people in the class who are not friends. This is not the same as what the original statement means; but you're very close to having the right answer.
Break it down:
Can you see how to put this together?