Help putting an Arduino project on the Internet
What would be the most efficient, easy to use and reliable system?


What seems to me is that you need what is called Unified Application Routing. There are various technologies to do that but what I suggest is WAMP  (Web Applicaton messaging protocol, do not get confused with wamp server stack) but you can check this comparison table and select the one that most helps you. Based on that what I suggest is crossbar.io. Is open source with a lot of examples, runs on small devices (raspberry pi for example) and is intended for the use you propose, connect devices and let users/apps to get/give information from them. You can publish/subcribe to topics and all devices or apps connected with that topic can read it. Imagine something like an arduino is reading temperatures from a room and publish that data to topic "Temp1". All devices(other arduino, servers, esp8266)  or apps(websites, programs, what ever you clients you imagine) that also are connected at that moment to topic "Temp1" will get that update and act like you want. Wamp also provides a nice feature that is called RPC (remote procedure call). Imagine that you want some real nasty thing in your arduino but it lacks memory or is too slow for that. You can create a program that is, lets say in a raspberry, just waiting for that calculation and returns the result to arduino.

Take a look at this video as an example of what you can do with this ttype of techonologies. A lot of demos also in crossbar.io page. The beauty is that you can program in almost any language you are familiar with. They rely on websockets for the majority of things but crossbar is not restricted to it only, you can even use raw sockets.

As for the second part.

If you create your on server (crossbar lets say) inside the lan you have to forward the port he is listening in your router so other outside your network can access it( and possibly will need to get some kind of Dynamic DNS service if your internet provider do not have statics IPs). Or you can host your server outside your network (crossbar has a tutorial for Amazon EC2 and for Microsoft Azure I think). You only need to port forward server ports.
Hwo do you connect arduino? Use ethernet shields, esp8266, choose a yun or make a combo like arduino with usb + raspberry. That way you can also use your raspberry as a crossbar server.  

Here is an example of my home made crossbar.io server using 2 python programs , 1 rboard and a raspberry pi has a host to manage my home heating system and the webpage is being accessed from internet, not local lan.