Basically it would be cool to play league on Mobile, but it cant not now at least so wouldnt it be cool if we had a menu style of the League on PC on Mobile? So you could buy RP on Mobile with iTunes Etc. And talk to friends from the APp and Look at whats happening etc Basically the Look of the Leageue Desktop Would be nice!
Playing league on a phone? For real bro... You can barely play angry birds without a crash/bug. Besides, many apps exist that tell you your stats and your friends stats. Try iLol I use it and it's perfectly fine. You can even chat
I'm not sure what your asking for.
There are MOBA's built for phones (namely Vain Glory), but League could not be adapted to that. The complexity of the games interactions would not able to be pulled off without a mouse.
Purchasing RP can be done via Amazon for NA at least (not sure on expansion plans). Riot would likely provide this service via their web store before they'd make an app for it.
There are apps out there that let you chat with the League client via your phone.
As for the look of it, if you're on Android you can download a different Launcher and customize it to look exactly like the client if you're willing and patient enough.
