Problem solved! Good folks at Elementary just posted an update to Camera. This Logitech model is now recognized by Elementary's Camera app and by Google Hangouts. Motion is actually rendered more smoothly in Hangouts.
I had the exact same problem, MacBook Air late 2014.
On Elementary Os Loki 4.1
after doing everything exactly the same as you, I ran:
sudo modprobe -r facetimehd
sudo modprobe -r bdc_pci
sudo modprobe facetimehd
And now the camera works perfectly.
It's here in "additional notes": https://github.com/patjak/bcwc_pcie/wiki/Get-Started#devvideo-...
Cheese and "camera" seems to work all of a sudden. I would suspect it was from a linux kernel update or Elementary that I think happened. If this happens to you, either run it as sudo or wait until another update comes.
AFAIK, there isn't, but I use cheese for my webcam.
sudo apt-get install cheese
It has many more options than the builtin camera. Yea, I would love to stick with the builtin stuff, but it needs many more features before it can be considered a full webcam program.