The answers you give are wrong. It is easy to verify that $n, n\log n \in o(n^2)$, and similarly $n^2 \log n, n^3 \in ω(n^2)$.
Give two functions that is in $O(n^2)$ but not in $o(n^2)$.
Looking at the definitions, we see that there are two ways to come up with such a function:
Pick any function from $\Theta(n^2)$.
Besides the obvious candidates of the form $cn^2 + o(n^2)$, there are also more obscure ones like, for instance $\bigr(2 + \frac{\sin(n)}{\log n}\bigl) \cdot n^2$.
Pick any function $f \in O(n^2)$ for which $\lim_{n \to \infty} f(n) \cdot n^{-2}$ does not exist.
Simple examples are $(2 + \sin(n)) \cdot n^2$ and
$\qquad \displaystyle f(n) = \begin{cases}n^2, &n \in 2\mathbb{N};\\n, &n \in 2\mathbb{N}+1.\end{cases}$
There are many more.
I'll leave the proofs to you as an exercise.