A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. $\sum_{k=1}^\infty a_{\sigma(k)}$ converges and $\sum_{k=1}^\infty a_k = \sum_{k=1}^\infty a_{\sigma(k)}$. For real valued series we know:
$$\sum_{k=1}^\infty a_k\text{ is unconditional convergent} \iff \sum_{k=1}^\infty a_k \text{ converges absolutely}$$
First Question: Is there also something like unconditional divergence for series studied in mathematics? It may be defined as
A series is $\sum_{k=1}^\infty a_k$ diverges unconditionally, iff $\sum_{k=1}^\infty a_k$ diverges and for each permutation $\sigma:\mathbb N\to\mathbb N$ also $\sum_{k=1}^\infty a_{\sigma(k)}$ diverges.
For example $\sum_{k=1}^\infty 1$ or $\sum_{k=1}^\infty k$ diverge unconditionally.
Second Question: If it is already studied in mathematics: What is the characteristic property of unconditional divergence for real valued series? It cannot be $\sum_{k=1}^\infty |a_k|=\infty$, because a divergent rearrangement of $\sum_{k=1}^\infty (-1)^k \tfrac 1k$ would be a counterexample...
An unconditionally divergent series in the sense you described is simply a series that tends to infinity. It's easy to see that a series whose partial sums tend to infinity is unconditionally divergent; conversely, a series which diverges but doesn't tend to infinity must oscillate, and this can be used to show that it can be made convergent by an appropriate permutation of its terms.
See On Divergent Series by A. S. Chessin, Bull. Amer. Math. Soc. Volume 2, Number 3 (1895), 72-75.