Comparing the big-$O$ of these four functions
One useful technique for comparing functions asymptotically is to take their logs since doing that turns exponentials into polynomials. So, if you are comfortable with comparing polynomials, then taking logs is always something you should be thinking about.
There is one thing that you have to be careful of when you do that, however. After taking logs, you can not throw away constant factors anymore.
If you take the logs of your functions, you get $\log(f_1(n))=\sqrt{n}\log(n)$, $\log(f_2(n))=n\log(2)$ and $\log(f_3(n))= 100\log(n)+\frac{n}{2}\log(2)$. So, $f_2$ is asymptotically larger than $f_3$ which, in turn is asymptotically larger than $f_1$. You already know where $f_4$ goes.