I'm pretty confident you hear the unlock sound when you open a locked door, but I'll verify.
These do seem to be old, simple locks in any case; we've skeleton keys. Doubt we have self-locking mechanisms.
Edit: From a sound perspective, you 100% unlock the door as you open it, and lock the door as you close it. It seems to make sense to me that the door isn't locked if you don't have a key. Hell, even 40-60 years ago you had to use keys on a lot of doors to open them to /leave/ a home if they were locked. Not just deadbolts (which these skeleton key locks would be similar to. They're either engaged, or not. Keyhole on both sides.)
Not a bug. Doors cannot be locked open (that doesn't even make sense if you think about it?). When a burrow owner opens a door, it is unlocked automatically to save them the hassle of doing it separately. Likewise when it is closed. Guests cannot lock your doors without a key so they do not lock it when they close the door. Austra is mistaken.