plotting frequency response
why is that when we plot H(w), we need to think of it as a complex exponential?


We don't need to think of it that way but it's convenient in this case to do so.

We know there's a property of Fourier transforms where if 

\$x(t) \iff X(j\omega)\$,

then

\$x(t-t_0) \iff X(j\omega)e^{j\omega{}t_0}\$

So in this case writing the frequency response in this form makes it clear that the time domain effect is a delay shift.


  But what if H(w) is not of the form \$e^{-j\omega{}t_0}\$, but it is in the form of \$j\omega\$. How can now we plot it? (It has no complex exponential on it so theres no way to regard for magnitude and phase.)


There's no reason it should have to have an exponential component. Any complex number has a magnitude and phase.

For example, 3 + 3 j has magnitude \$3\sqrt{2}\$, and phase \$\pi/4\$. And \$j\omega\$ has magnitude \$\omega\$ and phase \$\pi/2\$.

In general if you have a complex number a + jb, its magnitude is \$\sqrt{a^2+b^2}\$ and it's phase is \$\mathrm{atan}(b/a)\$.

If you have a complex response function that depends on frequency, you can always plot its magnitude and phase as a function of frequency.

If you already know the magnitude and phase of a complex number it's convenient to write that number in the form \$Ae^{j\phi}\$, but that's not the only way to write it. You can also write \$A\cos(\phi) + jA\sin(\phi)\$.