To access the security camera from the Internet, you just need to know the router's Internet IP address, what ports your camera software is listening on, and set up port forwarding. Forwarding is literally that - when the router receives an incoming request to the Internet/WAN IP on a specified port, it simply forwards that request to a specified internal/LAN address on a given port, which can be the same or different. Say the camera listens on port 3278, you forward that port to the camera's internal IP of 192.168.1.103 and then connect to the router's WAN IP on that port, 68.175.110.128:3278. You can mix up the external port if you want; that can be a good security measure given that a lot of malware will target known but obscure ports that may be opened by things like camera software. It's a product-specific process, but it's not terribly hard. Guides for most any router should show up in a quick search.
It can also be grossly unsafe since you're no longer in stealth mode, but that's another long explanation that I'll spare you unless you're interested. I'd recommend having the kids sit down and show you the whole process in great detail. The whole magical iPhone is making me very uneasy, TBH...opening up a router to the Net shouldn't ever be done automatically.
I'm thoroughly confused by what you're asking re. DNS though..."it doesn't work that way" is the best I can come up with for now. Passwords and host connections don't really have anything to do with it...here's the wiki, that might help, or at least clear things up so I can understand what you're asking.