I've read that followers can open locked doors and containers for me but when I order my follower Lydia to do so she responds with a negative. How do I get her to do this?

she won't do it because it's not in her nature . u need a shady companion to do that type of work . theives guild maybe ?

It worked just fine for me, even with adept locks. I don't expect lydia to open chests/doors you aren't allowed to enter though.

It was a normal "Adept" level lock in a dungeon. No key required and no "stealing" involved. Do I need to give her lockpicks?

I never gave her lockpicks. So I don't know the answer.