Hi there. Great question!
ERA is intended to work in a local network (LAN). So if you start it in a computer at home, all other computers connected to the same network can access all modules using the IP mentioned in the server dialog (or you can check the address in the browser). Also, if you have a WiFi router, all devices that connect to it can also access the modules.
Just as Soulcleave said, if your phone is using Data/H/Edge/3G/4G/etc... you are not really connected via your WiFi router, so you can't "see" the ERA server, since it's not out there with the rest of the Internet, it exists only locally.
There is a way to allow people to access your ERA server from outside, but you'd be entering non-safe usage of your home network.
What you need to do is called "opening a port". A good starting place might be here:
https://portforward.com/router.htm. For ERA, you need to open port
7777.
Now, please consider the safety of your home network when doing this. It should only be done for short periods of time.
I use this technique sometimes to allow my players to level up while they are at their homes, and once a player leveled up while on a bus on his way to the gaming session.
It is a great tool, but it requires a minimum of technical knowledge and can be dangerous if used without consideration.
To make something like this more widespread and useful, I would need to set up a public ERA server, hosting the different adventures for registered gaming groups. But that would cost money and time, for which I would have to charge monthly, and right now I don't think there are enough users of ERA as would be required to keep that architecture running.
About the server running after the server sleeps, all modules have a timeout function to avoid eating unnecessary memory/processor in the server. If you don't use a module for some time, it will stop responding. BUT if you just press F5/Refresh/Reload in your browser, it will get you a new session for the same module. The only module that will lose information when doing this is the character creation. It would also happen in the character sheet, but if you're not going to make changes for a while, you can just hit the Save-Server button every now and then to make sure the changes are saved on the server.
Does that answer all questions?
Thanks for the discussion!