Does anyone have any ideas or tips on how to create the backend for the games? Any useful framework, hosting service...? Thanks!

As for hosting service, I recommend Vultr, Linode or DO
If you don't want to deal with this headache, I suggest you to try modd.io (it takes care of backend & hosting completely)

I recommend Heroku which is free, i host on it one of my programs. Just make sure you dont use too big process type. Elsely you can use Cloud9 at c9.io or digital ocean.