In fact there are two different meaning for the term "global anomaly", which is a pity:
- global anomaly as opposed to gauge anomaly;
- global gauge anomaly as opposed to local gauge anomaly.
An anomaly can arise from global and gauge symmetries. So here global refers to the fact that the symmetry group is not gauged: these symmetries have a physical interpretation and are not spoiling the theory.
On the other hand a gauge anomaly arising for a gauge (or local) symmetry is really showing that the theory is inconsistent and should be fixed.
This explains the first point.
Concerning the second point there are two types of gauge transformations: infinitesimal (also called small) and finite (also called large) ones. The infinitesimal ones are the most studied, but some effects appear only when one considers finite transformation (for example to reach a component of the group not connected to the identity). Both kinds of gauge transformations can lead to an anomaly: a "global gauge anomaly" refers to a large transformation while a "local gauge anomaly" refers to an infinitesimal transformation. In particular it is possible that a naive analysis with infinitesimal transformations gives the impression that there are no anomalies, but one needs to look at large gauge transformations to be sure.
In both cases the discrete character of the group does not enter. Nonetheless let me answer to your questions. For 1) and 2) global and local anomalies can exist for both discrete and continuous symmetries. I am not familiar with this topic, but you may find more information in the following papers: arxiv:1404.3230, arxiv:1412.5148. For 3) the first paper discusses of discrete global anomalies.