First off, it applies to all elemental damage whether it's from a spell or attack. The penalties/bonuses do not stack with multiple hits. Chaos is not an element, fire, ice, and lightning are the only elements. The way multiple elements at once work is that the penalty is applied to all the elements you hit with on the previous attack, and the bonus to the ones you didn't hit with on the previous attack. So if you hit with both fire and ice, on your next attack fire and ice will BOTH have penalties, but lightning will get a bonus. If you hit with one just one element, only it gets penalized, while the other two both get bonuses. If you hit with all three, all three will be penalized on your next attack.
The penalties and bonuses last for 5 seconds, or until they are overwritten with a new elemental equilibrium strike (ref). When soloing each of your hits will reset the resists/bonuses based on what you hit with.
The changed resistances will apply to all damage dealt to the target, whether it comes from you or your allies. However, only hits from you or your traps/mines/totems will trigger the effect - so you could for example attack with lightning damage to apply the effect while your minions or party members attack with cold and fire damage to make use of the lowered resistances.