I thought I'd be nice and show the host of embers the Undead Settlement estus soup room (the one you fall down into in the mid-area room with lots of hanging bodies). But I can't open the door, even though most small doors can be opened as a phantom! Is this somehow intentional, a bug, or am I missing way of getting out (other than suicide or the Black Separation Crystal)?
This is seems to be intended. Phantoms can't open doors (or at least they can't open one-way doors. This might be to prevent an experienced phantom from opening a useful shortcut). Your ways out would be:
Have the Host of Embers notice all that delicious soup, so he comes down to taste it and opens the door himself
Use the Black Separation Crystal to return to your world
Use a spell that kills yourself (if you have any) to die and return to your world