What is the hotel room card slot inside the room for?

I have seen these cards control

• everything including plugs (so annoying to leave your laptop charging while you go out for the day, and come back to find the plug was not on while you were gone)
• just lights but not plugs
• lights and air conditioner but not plugs
• just the air conditioner

Whether it exists and what it controls seems to vary around the world. In Europe it's most likely to be lights, or lights and plugs. In the South Pacific it was usually just the airconditioner (and there was a sensor on the door to the balcony or deck that would also turn off the ac) or the ac and lights. In Europe the staff generally didn't tell me what it was, in the South Pacific they generally did. I have seen these occasionally in North America.

It is easy to fool using either your second key, a card from your wallet you don't need right now (grocery loyalty card say) or a folded up bit of card like a magazine subscription card. Some hotels don't care if you do this, but in my experience, in hot places where it's hooked to the ac they care. One place said there was a $50 fine if they found the ac on when you weren't in the room. It is all about the cost of cooling and lighting a room you're not in.

While the other answers have elaborated on the primary reasons, I am going to point out what I perceive as a secondary reason and/or benefit:

The key has its fixed place.

To me as a hotel guest, this seems incredibly beneficial because it means the key does not lie around somewhere where I might forget it when leaving the room. Instead, it's in its fixed place in the power slot, where I will invariably look before closing the room door. If the key is still there, I will notice it (and thereby spare myself of an embarrassing request at the reception desk to help a guest who was stupid enough to look himself out), and if it is not there, I know I have taken it.

Note that by disabling the lights when the key is not in the slot, the chance that I have put the key somewhere else upon arriving at the room the last time is close to zero.

Of course, this doesn't work quite as nicely in a situation you describe:

though I've got the standard 2 cards for my room so I have another one anyways

Personally, I have stayed in hotels in various countries and never got two keys, unless I was there with my spouse, in which case each of use got one key and the method described above works again.

EDIT: This method works for any design of the "power card", be it a credit card-sized key card, or rather the keyring pendant for a physical key, like this, or this, which may not easily fit into a wallet.

The 15 minutes problem is most likely caused by a faulty motion detector. Many hotels use a motion detector to switch off the main lights when the guest is sleeping. You can sometime notice this after waking up, sometimes the motion detector has not been triggered and then the lights don't immediately work.

Most often a circuit breaker to cut or provide power to the room and/or to unlock the door. That's generally fairly obvious, so if not, It's possibly deprecated and bypassed. When being used as such it is typically just a power saving device. Imagine the annual cost saved, especially by large hotel chains.