Chess Engines, Elo Rating, and Depth Level?

I am playing with Lucas Chess but also use Fritz and Chess King.  These programs allow you to adjust the depth level (how many moves the engine looks forward).  In many instances, the engine also has an assigned Elo rating.  What I can't figure out, is if you're playing an opponent with an Elo of, say, 1400 and then increase the depth level, wouldn't the Elo increase?  Wouldn't the opponent be more difficult?

For example, in Lucas Chess, there are various opponents assigned Elo ratings.  You can click on an opponent.  By default, every opponent is set to 3 moves depth.  This doesn't seem to make sense to me.  I would think tougher opponents would have higher depth, not a default level of 3, which is what I find on Lucas Chess.

Is it the engine, the depth level, etc. what determines the level of play by the computer?  How come all of the Lucas opponents show a default depth level of 3? 

I'm making some assumptions here, because I'm not familiar with the specific programming of these engines. That said, I assume that each opponent would use a decision tree corresponding to the depth setting, and the Elo rating determines which combination the opponent chooses from that decision tree. At 3 moves ahead, there is a tremendously large set of potential combinations. If you bump up your opponent to 1400 vs. 1200, for instance, perhaps the engine will choose a more powerful combination that that the 1200 Elo opponent would miss. You're right, though, that this is confusing, as there is definitely significant correlation between depth and Elo, so adjusting one in reality would impact the other. I think this confusion results from the difficulty of creating a chess engine which actually mimics the mind of players of different Elo ratings.