I informed my employer. The Human resources person was pretty surprised. I think I am probably the first one, and probably the last to have done this. I know quite a few other employees at my company have naturalized. They took a copy of my passport. They took copy of GC when I was on GC, so basically my company makes copies of documents presented. I know they don't need to do this, but they do, and I don't care much.
Yes, as far as I know if you are on GC, you can obtain employment just by showing driver license and social security card and they shouldn't bother to ask you again to prove your status, as permanent residents have (as the name clearly say) permission to reside and work in the US permanently. The GC expires after ten years, but even if you let your GC expire without renewing it you are still a permanent resident. Of course, in my case I naturalized way before my GC was going to expire. Considering how long USCIS takes to renew a GC I don't think USCIS or ICE is going to enforce the law of carrying the GC with you at all times very strongly. I would assume every year there must be hundreds of thousands of permanent residents waiting for their new GC.