Some (maybe most?) email programs (Thunderbird for example is free) have a local folders section in the folder list. Just create your archive folder in there.

For the phone, it should have a setting to only keep the last number of days on it. Everything that happens will still get to it, but the phone will remove anything older automatically. Some phones should also let you choose which folders to watch. You might have it set for "all".

Of course, make sure that the local folder is located somewhere so it is included in your backup routine.