Is my intuition correct here?
If our metric space consists of a single point $x$, any sequence in it would just be $x, x, x, \dots$ which is obviously Cauchy.
If we have any metric space $(M, d)$ with at least two points $x_{1}, x_{2}$, we can define the sequence $x_{1}, x_{2}, x_{1}, x_{2}, \dots$. By the properties of a metric, $d(x_{1}, x_{2}) = a > 0$. Thus, for any $0 < \epsilon < a$ and any $N$, we note $d(x_{N+1}, x_{N+2}) > \epsilon$, so this sequence cannot be Cauchy. Thus, no such nontrivial metric space exists.
Yes. That's indeed right and quite easy. There are quite a few non-trivial spaces where every sequence has a Cauchy subsequence (this characterises totally bounded spaces, IIRC), though.