Wi Fi and DNS with Arduino Mega

I m developing a project with Arduino Mega for elder people. This is a carry on device, so the device will be moving in different wifi networks with obviously different ssid name and different password. The device will have a display and a query will be displayed requesting these two information. On this times any one can help with this configuration and the issue is not big deal. However as the data acquired by the device must be monitoring periodically with a system installed in a computer managed by relative of the person or a doctor (could be in a different city or country) we need to know how to connect by internet to the Arduino. No many people know how to get the dynamic IP assigned by internet to a device so we need to use a NAME instead an IP. Some questions pup up…

1.- What king of wifi device we need to connect to the Arduino Mega to support the programs, routines… whatever needed to fullfil the requirements
2.- Do I need to access or create a DNS server to be able to use a name instead an IP? If yes, how can I do this?
3.- How to setup the wifi device and how to setup the arduino?

Thank you in advance.