Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian?
NOTE:
I've read the derivation but sometimes i see it's like a definition or a claim in the case of DTFS.
This is, essentially, what the sampling theorem is about. Uniform sampling in one domain (e.g. the time domain) causes periodic extension in the reciprocal domain (e.g. frequency domain).
The reason why is that the sampling function is a periodic function which means it can be represented as a Fourier series
$$\begin{align} \mathbf{Ш}_T(t) \ &\triangleq\ \sum_{k=-\infty}^{\infty} \delta(t - k T) \\ &= \sum_{n=-\infty}^{+\infty} c_n e^{j 2 \pi n \frac{t}{T}} \\ \end{align}$$
where the Fourier coefficients are
$$ \begin{align} c_n\, & = \frac{1}{T} \int_{t_0}^{t_0 + T} \mathbf{Ш}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \quad ( -\infty < t_0 < +\infty ) \\[4pt] & = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \mathbf{Ш}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\[4pt] & = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\ & = \frac{1}{T} e^{-j 2 \pi n \frac{0}{T}} \\[4pt] & = \frac{1}{T} \ . \end{align} $$
Using the definition of Fourier Transform most common with electrical engineering
$$ X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ \mathrm{d}t $$ $$ x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ \mathrm{d}f $$
when sampling $x(t)$
$$\begin{align} x_\text{s}(t) &\triangleq x(t) \cdot \big(T \cdot \mathbf{Ш}_T(t)\big) \\ &= x(t) \cdot T \sum_{k=-\infty}^{\infty} \delta(t - k T) \\ &= T \sum_{k=-\infty}^{\infty} x(t) \ \delta(t - k T) \\ &= T \sum_{k=-\infty}^{\infty} x(kT) \ \delta(t - k T) \\ \end{align}$$
which shows how $x(t)$ is converted to samples $x(kT)$ and this is also true:
$$\begin{align} x_\text{s}(t) &\triangleq x(t) \cdot \big(T \cdot \mathbf{Ш}_T(t)\big) \\ &= x(t) \cdot T \cdot \sum_{n=-\infty}^{+\infty} \frac1T e^{j 2 \pi n \frac{t}{T}} \\ &= \sum_{n=-\infty}^{+\infty} x(t) \ e^{j 2 \pi n \frac{t}{T}} \\ \end{align}$$
the resulting Fourier transform is
$$ X_\text{s}(f) = \sum_{n=-\infty}^{+\infty} X\left(f - n\tfrac{1}{T} \right) $$
which is periodic in the frequency domain with period $\frac{1}{T}$.
Because of the Duality of the Fourier Transform, if can be also shown that sampling in the frequency domain causes periodicity in the time domain, which is essentially all that Fourier series is about.
Regarding your first statement, discreteness in time or frequency domain does not always dictate periodicity in the other domain. Only certain kinds of discrete sets do. For instance, a set of impulses that are spaced apart by relatively irrational values will not be periodic in the other domain.
