I want to increase the range of my router on the upper floor, for gaming in another room. For this, the cable will have to run alongside metal stair railing. Will this cause interference? Will the use of STP/Coax cable reduce this interference? Secondly, the cables are gonna run outside the wall below electric sockets(about 24 cm away), will there be any problems due to this?
Metal parts of the structure, such as stair railings, don't generate any interference at all. So there's nothing to worry about there.
Running close to the power cords won't cause any trouble at all. Long runs that are very close to cords / cables carrying high current may cause trouble, but not the setup you're describing.