I've not used peer to peer replication, I think it would do what you want (active-active servers) but it's an Enterprise Edition feature.
Merge replication would also do it. Data changes could happen on either server and if one went down the changes would be queued up until it came back (up to a configurable amount of time).
If the server had to be rebuilt then you'd have to set the replication up again from scratch or restore backups if you have the replication scripted and suitable backups.
Transactional replication is one way so if you lost the primary (publisher) you wouldn't easily be able to get the data back to it when it came back.
If you are just after having one server that is active and a secondary server that is inactive and takes over in the event of a problem (and can be switched back and forth) then mirroring, AlwaysOn Availability Groups or Log Shipping might be of use.
Regarding the folders, the replication agents runs via executables that are called by SQL jobs. The agents run under Windows accounts that are specified when replication is set up and those accounts need read/write on the snapshot folders. Do you get any errors showing up in the SQL jobs?