I started by removing my gear to get best controlled answer possible to make my damage not fluctuate so much. I equipped a low level bow to my demon hunter and went to hell act one. There I started my tests on zombies using my elemental arrow since it does the same damage under different element types due to runes. Against the zombies I use Elemental Arrow with no runes as a basic constant. It dealt around 200-300 damage each shot. I then changed the runes to Frost Arrow and electric Arrow. The outcome was the same 200-300 damage. I decided to test my idea on a more diverse group of enemies. I found some carrions and moon clan, repeated the experiment and had the exact same outcome.
I decided to use test on a creature that I was positive had some type of resistances, Inferno Zombies (which are on fire) in the Halls of Agony level 2 in Act 1. I shot the monsters with Fire and it averaged around 160-250 a shot. Electric averaged 200-300, and frost arrow averaged 200-300. I then decided to try to see how this scaled with gear. I equipped some of my gear and went back in to the area. My Fire arrows were averaging around 1,800 while my frost arrows were averaging around 2,800.
As A final test I decided to go to act 2 and hunt down a construct. I found a Chilling Construct and a Burning Construct and shot basic Elemental Arrow (which is fire damage) and averaged for around 2600 damage on the chilling but 800 on the burning. Swapped to the Frost arrow rune and averaged around 1000 damage on the chilling and 2400 on the burning. Then swapped to the Electric Arrow and averaged around 2500 damage on both constructs.
From the test I have gathered that an enemy has a high resistance to the element that it is, but will not have lower resistances to the opposite elements of its type. Also, an easy way to tell what element type a creature is would be by the type of damage it is dealing (example the Constructs stated above.)
From the standpoint of your weapon: Weapon Damage is king. Above all else, you want the highest possible amount of raw damage and DPS possible. Beyond that, you'll want to take a look at this formula to understand the value of other stats.
In short: at low levels, nothing is going to beat raw +Damage on any slot where you can get it (Rings, Neck, Offhand). As you increase in level, you'll eventually reach a point of inflection where the multiplicative scaling gained from adding +Dex is more valuable than the straight + Damage. This point of inflection will vary with other stats, and particularly with the base DPS of your weapon - your best bet is to simply keep an eye on the +Damage portion of the tooltip when comparing items.
As for stats like Crit and Attack Speed, the same basic rules apply. In general, .1 attack speed is worth 10 Dex, but this ignores the defensive benefits of Dexterity, and since attack speed bonuses are provided as percentage multipliers, you'll need to do some math. You'll need to run similar calculations with +Crit and +Crit Damage, but in general, unless you use skills or passives that rely on Crit, you can assume those stats are less valuable than the others.
Really, all of this comes back to one key point: Trust the Damage Stat on your Character Sheet. So long as you're doing the basics right (using a level appropriate, high DPS weapon), it isn't going to steer you far wrong. Beyond that, Dexterity is King, and everything else depends entirely on the magnitude of the stat that you get. An item with +Crit might be better than an item with +Attack Speed, based purely on how much of it is there.
Best Answer
This is a mixed bag.
If your weapon element matches the bonus elemental damage, OR if it is a black weapon. The bonus is added to your attack.
If your weapon element does NOT match, you will not gain the bonus when using a non-elemental attack (for instance, using a fire weapon and casting magic missile will not give you the added +% lightning bonus). This is mostly true but remember that weapons have a bit of base black damage so that may be added to the skill (you'll see a small up tick in DPS on your character).
If your weapon element does NOT match, BUT you use a skill that converts your damage to that element, you will get the % bonus for that skill attack. (i.e. casting shock pulse while wielding a fire weapon gets you the % lightning bonus).