If the sum of the tail of a series goes to $0$, must the series converge?

The result you want is obvious. If $$\lim_{m\to\infty } \sum_{n=m}^{\infty} a_n = 0$$ then for some $M$ we have that $$\sum_{n=M}^{\infty} a_n <\infty.$$ Tacking on the first $M$ terms doesn't affect convergence.

Yes, it is correct also when the sequence $(a_n)_n$ is not positive because $(S_n)_n$ is a Cauchy sequence $$|S_n - S_m| = | \sum_{k=m+1}^{n} a_k|=|\sum_{k=m+1}^{\infty} a_k-\sum_{k=n+1}^{\infty} a_k|\to 0$$ as $n,m\to \infty$.

Here is D. Brogan's and Mohammad Riazi-Kermani's proof written out with all the details I think are necessary.

Since $\displaystyle\lim_{m\to\infty}\sum_{n=m}^\infty a_n=0$, there is an $M$ such that $\displaystyle\sum_{n=M}^\infty a_n$ is finite, say it's equal to $\tilde L$. For $k\geq M$ let $\displaystyle\tilde S_k=\sum_{n=M}^k a_n$. Then we have, for $k\geq M$, $\displaystyle S_k=\sum_{n=1}^{M-1}a_n+\tilde S_k$. The right hand side is a sum of a constant and a convergent sequence, hence $S_k$ is also a convergent sequence, i.e. the original sum converges.