Although it's the same socket, the fact that it's a "random HP board" as you say, doesn't necessarily mean it's supported with the board, especially if it's from a random pre-built.
You can certainly use the r9 270, it's just it will draw more power even if idling. More power means higher running costs.. And of course heat, but if it's a machine you only intend on turning on when you or your friends are online then that won't really matter.
Also you mention there is no internet in your room, does this mean you intend on using a Wi-Fi card in the server or placing the server near wired access? I would certainly suggest the latter.