As per the title, are mods (and non-mods with semi-mod powers due to high privileges) notified and/or allowed to moderate their own posts which others have flagged?
Only actual ♦ moderators can see a summary of all flags and act on these (i.e. dismiss as helpful or decline them).
Users with 10k privileges see a subset of flags and may act on them by flagging the posts another time or disputing the flags. They cannot, however, act on these flags in any other way. As a consequence, "normal" users with 10k reputation cannot see flags on their own posts and can therefore not act on them.
Since moderators see all flags, they also see flags on their own posts. This means they could also act on these flags, although I guess many moderators would leave flags on their own posts to be handled by another moderator for the sake of fairness and transparency.