I think the elements multiply tha damage by the amount in the corner, like x3.
Fire works effectively on enemies without shields. Shock has a chance to instantly deplete shields. Corrosive works well on enemies with shields. Explosive makes bullets stronger with no added effect.
I'd stick with the elemental gun, unless the damage of the non elemental gun is higher than the damage of the elemental gun times the multiplier.
You can use any gun you want, and build profficiencies in anything. The perks of using guns that the classes are made for are skills that can help them with those weapons. For instance, Mordecai gets skills that improve the effectiveness of sniper rifles and revolvers. But I prefer to use snipers, shotties and combat rifles.