Hey,
Yes, if a cheap PSU dies it can fry your components too. A good PSU is essentially the backbone of the system, without it your computer might not be stable or could get damaged.
As for other components, if you have a decent CPU then the most important thing for gaming would be the graphics card as it affects game performance the most. As for RAM it doesn't really matter, but try to get 1600mhz DDR3 RAM. RAM prices are quite low so it should be affordable. However you don't really need anything more than 1600mhz as it barely affects your performance in games.
Hope this helps!