When a user rapidly clicks the sleep/wake button five times, two extra functions are offered: medical ID and emergency SOS:
Can third party developers create an "emergency" app to appear in this list?
I don't think that you can setup emergency apps other than the two mentioned. Basically you can quickly call an emergency number and after the call you can decide to not notify the previously added contacts, if any, by text message. Contacts are also notified when you location changes.
This feature is described here:
https://support.apple.com/en-us/HT208076
Excerpt:
"With Emergency SOS in iOS 11, you can quickly and easily call for help and alert your emergency contacts."
and here for the Apple Watch:
https://support.apple.com/en-us/HT206983
Excerpt:
"With Emergency SOS, you can quickly and easily call for help and alert your emergency contacts."