Thought I'd just post an update on the introductions here rather than starting a new thread (mostly since Claire has so many already!).
Anyway, I'd never tried Feliway diffusers before, but got a couple on and plugged them in over the weekend - expensive, but worth it if it makes things easier for my crew. Over the weekend I started letting Claire have the full run of the house when I was home. She was a little hesitant to venture out and was quick to retreat to the spare bedroom if something startled her, but seems curious about her new surroundings. Peter's been totally relaxed with her. Katie's been keeping an eye on her from a distance or up above. Gracie's been following her around and growling occasionally, but hasn't been aggressive at all.
Since the weekend went well, I left the door to spare bedroom open when I went to work yesterday, leaving the cats alone for 9 hours. When I got home late yesterday afternoon everyone seemed calm and happy. Claire was in the spare bedroom, but lounging in chair, not hiding the closest. It's obviously the place where she feels the most secure. I can't imagine how scared and confused she's been for the past couple of weeks, losing both her home and her person. She's been pretty playful and has been coming to me for attention - both good signs.
I've decided to keep Claire confined in the spare bedroom at night for at least another week. I don't want to push it too much with Katie and Gracie.
If there's any signs of stress, we'll slow things down, but at this point, I'm feeling pretty optimistic that everyone will at least tolerate each other. I think they know by now that there's enough love to go around!