Are there any situations in which permanent residents ARE allowed to
vote (local elections, ballot initiatives, primary elections, etc)?
Some special districts (e.g. a ditch association) or proposed special districts (including, e.g. annexation petitions) afford a right to vote based upon property ownership in addition to, or instead of, based upon residency. In those cases, a green card holder who owned property in the district would probably have a right to vote, just as corporations who own property in the district or proposed district can.
Are permanent residents allowed to join political parties?
Usually, you affiliate with the political party by registering to vote, but nothing prohibits a political party from offering non-voter memberships the way that any other organization might, to minors and non-citizens.
Are permanent residents allowed to participate in political campaigns
as donors/volunteers/paid staffers?
Generally speaking, yes, but there are some exceptions. For example, Colorado's election laws provide that signatures of petitions for state ballot issues and to put candidates on the ballot may only be circulated by registered voters, which would exclude green card holders. So, in that case, some paid positions would not be available.