Mathematical advantages of the ZT, DTFT and DT?
My friend, i hope you can gain a proper command of those techniques for achieving an excelent result at your exam. Of course the question is broad and more discussionfull, but the purpose of this is "clarify" not to turn this into a "closary" of questions.

Indeed. there are a lot of variants for the transforms, but only a few of them have a clear direct meaning, and the others are generalizations seldom used.

At first, the standard, classical FT, is the main entry for your analysis. 
$$\mathbf{F}=X(f)=\int_{-\infty}^{\infty}x(t)e^{-i2 \pi ft}dt$$
$$\mathbf{F}^{-1}=x(t)=\int_{-\infty}^{\infty}X(f)e^{i2 \pi ft}df$$
That is the base, plain and simple, sweet, the FT and the IFT are the same (of course exchanging the signs and variables :) ) and with the easiest expressions $\mathbf{F}\{e^{-\pi t^2}\}=e^{-\pi f^2}$, $\mathbf{F}\{1\}=\delta(f)$, $\mathbf{F}\{\cos(\pi t)\}=\frac{1}{2}\delta(f-1)+\frac{1}{2}\delta(f+1)$, $\mathbf{F}\{\Delta(t)\}=\Pi(f)$, $\mathbf{F}\{\Pi(t)\}=\Delta(f)$, $\mathbf{F}\{|||(t)\}=|||(f)$ (where $\Delta$, $\Pi$ and $|||$ are the triangle, step and comb functions:)), and so on.
You use this on the handwritting, papers, reports, analysis, etc. 

We can safely ommit the DTFT and DFFT, and the tricky $i$ and $k$ symbols for discrete time and frequency and keep $t$ and $f$, discretized, with the sampling time $dt$  and the frequency periodic spacing $df$ (again with the same notation as the integral). Thus for the DTFT direct transform is:
$$\mathbf{F}_{DT}=\sum_{t=-\infty}^{\infty}x(t)e^{-i2\pi ft}dt$$
and the DFFT inverse is:
$$\mathbf{F}^{-1}_{DF}=\sum_{f=-\infty}^{\infty}X(f)e^{i2\pi ft}df$$

because the $|||(t)$ (and $|||(f)$) comb function:
$$|||(t)=\sum_{i=-\infty}^{\infty}\delta(t-i)$$

is all what we need for recovering the related expressions and cumbersome hard to learn theorems regarding sampling (DT) and perioditizicing (DF). And you recover all your symmetries just by multiplying by your combs:

$$\mathbf{F}_{DT}(x(t))=\mathbf{F}(x(t)\frac{1}{dt}|||(\frac{t}{dt}))=X(f)\frac{1}{dt}\frac{1}{F}|||(\frac{f}{F}), F=\frac{1}{dt}$$

Hence, at my second place, you have your well forgotten ZT, unilateral, again considering the $t$ as a discretized variable under a sampling time $dt$:
$$X(z)=\sum_{t=0}^\infty x(t)z^{-t}$$

You have a DAQ with a lot of sensors and controllers, and you have some fixed sampling time and you need to know the continuous parameters of the system based on the discrete behaviour, or tuning some controllers, or identifying a system. Soon or later you need to pass onto the Z domain and check if your IIR filters are stable or nor, or if your FIR smoothers are too short on delay, and comparing your ARMA models with your differential equations.