I studied covert, practiced it, perfected it, and I still can't open or close a door without coming out from hiding. It appears that I can open a container or close it without coming out from hiding though.

I've answered this before, but covert only prevents the 'so and so opened/closed x' echo (whether you are hiding or not). It doesn't stop you from stepping out of cover. For that, you need improved hide. (See help 'improved hide').