Most doors already have that script on them actually, even outside of dungeons Darkside of Heaven. Anything you have to open, unlock, pick up will incur a hide and Ms penalty. The reason you dont see the stealth checks is bc you're not hostile to someone/thing in your detection radius. The difference is that if you fail one of those checks, the ones you see in your combat log, you're completely destealthed instead of just being revealed to the creature that detected you.
I've spent a fair amount of time exploring our Stealth system, more than most I'd hazard. I like some of the ideas proposed but disagree on a few points. While it would be cool to not have the door open actually open, it's just not realistic. That door is open. It should be able to been seen as such and used as such. What if that door was trapped? What if someone was trying to follow you didn't notice? No amount of your own hide skill can ever make that door appear as anything less than open and it should be represented properly ig.
What I think you could do however, is a dex or Ms check to open the door silently. Here, the actual audio that plays when you open a door, chest, object, etc would be silenced. A bit like silent spell. Normally, the audio plays through out the entire zone. This is a bit unrealistic Imo. For example, if I'm at the palace and someone exits the bank, I can hear that door opening and closing despite being no where remotely near detection range. Observant players will still be able to notice a door opening/closing if they are in visual radius this way. I'm more partial to a dex check bc there will still be some chance of failure with normal dc. With an Ms skill check the chance of failing whatever the DC happened to be is virtually nil. Even buffed at Max levels a PC could still easily fail a dex check of 15.