You can run Webserver on Android for example :
KSWEB
but, there is 2 Case for use with Android Device :
First :
your phone Broadband is how much ?
this Broadband can support users
and online at 24 hours of day ?
second :
this is not good idea for start webhosting
or other something with a Android device.your hardware need backup and set other Options...
and about Questions :
1.
Point me at what to read about this, so I can set it myself
yes you can.use below link :
http://www.androidguys.com/2014/04/22/host-website-android-device/
2.
Tell me if the phone can run only on a charger, without a battery
(because I expect it won't live long if the battery is constantly
charging)
i have no idea, But this links maybe can help you :
http://www.makeuseof.com/answers/can-i-directly-power-an-android-device-by-circumventing-its-battery/
www.youtube.com/watch?v=Dp_sJYQR1Bg