Some hints to get you started:
The entire problem is about comparing the rates of growth of $n^n$ and $2^{2^n}$: do they grow at roughly the same rate, does $n^n$ grow a lot faster than $2^{2^n}$, or does $n^n$ grow a lot slower than $2^{2^n}$?
One way to investigate this is to consider the limit of their ratio as $n$ increases, $\lim_{n\to\infty}\frac{n^n}{2^{2^n}}\;.\tag{1}$ This is a bit messy to think about in this form, but you could write it in a more manageable form by using the fact that $n=2^{\lg n}$, where $\lg n=\log_2 n$. Warning: There are three $n$’s in $(1)$, and only one of them should be subjected to this substitution.
Once you have $(1)$, you can use it to address the technicalities of showing which of $\Theta,O$, and $\Omega$ applies in each part of the problem.