Glad to hear you’re underway with some Raspberry Pi projects! Re power, the Pi has a special way to turn it on if it were shut down properly (via command line or otherwise). Google “power on raspberry pi after shutdown”. It’s a bit finicky and to be honest, it’s easier to shutdown the Pi and instead use something like this to remove power / turn it back on http://core-electronics.com.au/usb-cable-with-switch.html
With that said, plenty of people live on the edge and just pull the power cord!
Re browser support, just google “openelec browsers” and checkout some third party options in that space.