$\sum (-1)^n b_n$ is representative of an alternating series.
We look at whether $\sum b_n$ converges and if $b_{n+1}<b_n$ $\forall n\in \mathbb{Z}$.
What if our alternating series is of the form $\sum z^n b_n$ where $z$ is any primitive root of unity. Do the same tests still apply?
Another question: $\sum 1/n$ diverges but $\sum (-1)^n 1/n$ converges.
Is there a $p \in \mathbb{Z}$ where $\sum z^n 1/n$ converges where $z$ is a primitive $p^{th}$ root of unity,
but diverges when $z$ is a primitive $(p+1)^{th}$ root of unity
In Dirichlet's test you can replace $(-1)^n$ in the alternating series test by any sequence with bounded partial sums, and thus as a special case also by $z^n$ where $z$ is a root of unity.
Thus the answers to your questions are "yes" and "no", respectively.