I once ran an errand that required milk, but didn't have a cow at home, so I just rudely broke into a stranger's farm and milked their cow. :D It worked with no problem, and I think it would work with collecting eggs, too. Then again, nobody actually caught me.
The only thing I know for sure is that you can't do it on public lots. I set up a store for animal treats and upgrade parts with a cow called "Gratis Milk" in the backyard, so self-sustaining sims without their own cows could harvest milk and never have to resort to what that previously mentioned errand boy did. Unfortunately, while it is possible to interact with her normally, I can't milk her.
