Is it unsafe to put router into "DHCP and NAT" mode? When I was trying to get my Raspberry Pi online, in airport utility I put my router in DHCP and NAT mode and entered ports (80). Did I open up these ports to the internet? Would this be unsafe to do? 

If I did not open up these ports to the internet, how can I open them up? 
(just to know what I should not do).
Yes, by putting it in DHCP mode you did open up your router to the outside world. Assuming you have changed the default password on your router then when someone tried to connect to your IP Address then it would simply send them to your routers webpage. In theory you would be safe from 5 year olds but it is not recommended.
Just for some extra information you would only want to open Port 80, Enable DHCP Mode or enable Virtual server if you were planning to host a server from inside your network that you wanted access to from the outside.
Hope this helped!