You need the .so's on the replica's simply for the database structure, but afaik, it's only the on-disk format that's being synchronized, functions are not executed on the replicas. I'm not even sure this would throw errors if the .so's would be missing - but it will give problems when a replica is promoted to the new master.
Yes.
Note if those functions have an implementation compiled from C, you do need to install the .so on the standbys though.