Most cameras record everything you're asking for, and much more. Canon DSLRs encode date/time, camera model, lens used, shutter speed, aperture, ISO, focal length, active autofocus sensors at time of exposure, and ALL of the camera settings. I'd be surprised if your back didn't as well. But that doesn't mean Adobe knows how to decode the manufacturer's proprietary method for encoding that data. If the software that came with the back can display the data, it's in the file and Bridge just doesn't know how to read it yet. Make sure you've downloaded and installed the latest version of Camera RAW, that may fix the problem.
But you don't want to try to edit the RAW after-the-fact to add the info. Attempting to do so may corrupt the file and render it unreadable.