Frasier could have forgotten his key int he earlier scene - locked it int he car or something. Which would have made a funny scene in itself, so it's sad they didn't include htat.
Lilith is Jewish - as we learn from the episode w/the circumcision. Since she is strict enough to require that, perhaps something in her background makes her upset that the couple was living together before marriage. It is still rather common for people to only live together after marriage, and would have been even more so in the late 1980s.
If Frasier had a key to lock the door at the end of the episode, then why did he use scissors/cheese knive to break open the door?
Lilith runs into the bathroom upset for the second time upon hearing that Frasier and Diane used to live together. She knew that they were engaged, wouldn't it be natural to assume that they were already living together by then?
Diane: Why is it that we cannot go to one single solitary place in the entire city of Boston without running into some woman that you've had a fling with?
Sam: Oh come on. You exaggerating here.
Diane: At the market. At the movies. When we got our blood test for heaven's sake.
Diane: May I propose another toast?
Sam: Why stop at seven?
Diane: Our relationship was a two way street.
Frasier: And I was run over in two directions.
Frasier: Dear, I thought I told you that Diane and I used to date.
Lilith: Yes, that you had decided to marry and spend the rest of your life with her, no.
Lilith: Why were you afraid to tell me that detail? Because she still means something to you, doesn't she?
Frasier: She means absolutely nothing to me. If I notice her at all it's merely as an annoyance like a bug flying around in a room.
Diane: You know what I love about your relationship?
Diane and Sam: We don't want to spend an evening alone with your kids.
Carla: Neither do I.
Lilith: Well a week ago we decided to take a chance on cohabitation. But we waited a week before announcing the news so as to avoid any undue pressure on the success or failure of the endeavour and now we are pleased to announce the effort a limited success.
Frasier: Doctors Sternin and Crane are proud to announce they are now officially POSSLQs.
Lilith: It means Persons of the Opposite Sex Sharing Living Quarters.
Cliff: Obviously [Carla] doesn't think that I can handle the kids. But my guess is that they'll settle down once they're confronted with a strong male authority figure.
Norm: Oh, you're bringing your Mom?
Carla: (looking for a baby-sitter) Hey, hey, what're you doing Wednesday night?
Al: Pumping iron!
A season one episode of Frasier was also called Dinner at Eight.