There are many ways we moderators can interact with questions where we are involved ourselves, either because we asked them, answered them or commented on any post in it. I don't think we should prohibit moderators from performing any moderator action in those questions. In most cases there is no major conflict of interest and we would reduce the reaction time of the moderator team if we would exclude the participating mods from acting.
There are several actions that moderators routinely perform that might present a conflict of interest:
You might construct a conflict of interest if a moderator has answered themselves, but I think that most of these cases are clear enough that there is no reason to prohibit moderators from acting.
When an extended discussion starts in comments, moderators often need to remove some or all of the comments. As long as the comments are not direct criticism of the mods question or answer, I see no conflict of interest. I agree that a moderator should not delete comments that critizice his posts, though the exception is anything that goes into offensive territory. If someone starts throwing around insults, any moderator, no matter if directly involved, should delete those.
I agree that moderators should not reopen their own question. At a point where a moderator and the community disagree to this extent, a meta question is probably warranted in any case.
I don't think there should be any restrictions here. If a user insults a moderator to the extent that a suspension is warranted, I see no problem with the moderator performing the suspension themselves. Every moderator (as well as the community team) is notified when a user is suspended, so there is a lot of review there anyway. Suspensions are also usually discussed internally before.