If I attach a crop lens (e.g., a Sony 50mm 1.8 OSS for Sony E-mount) on a full frame sensor (e.g., on a Sony A7), are there changes to the focal length, aperture, and F number? And will we have a shallower depth of field?
Focal length aperture and f-number are properties of the lens and remain the same, as always.
What will happen is you will get severe vignetting and the corners will be pure black. If you crop the central APS-C portion of the image you will get exactly the same result as on a cropped camera (except probably different resolution).
You'll probably be able to make a larger crop and still avoid the vignetting, so if you crop the middle three quarters of the image then you'll get some of the benefits of a larger sensor, producing a result equivalent to an APS-H sensor (1.3x crop), so field of view and DOF will be equivalent to a 65mm f/2.34 lens on a crop camera.
Edit: as the Sony A7 does not have a mirror, I believe Matt Grum's answer to be complete. My answer below pertains to SLR cameras.
You should check the flange focal distance. In some crop lenses, this distance is shorter and the rearmost lens element is therefore nearer to the sensor. Thus a full-frame mirror may collide with the lens element.
I know this applies to some Canon lenses, but you might have problems with some wide angle lenses.