Aren't brining and dry-aging contradictory?

Fat distribution in the cuts you're thinking about will give you your answer.

Poultry is lean, a good steak much less so.

It's not always that simple. Curing ham or pastrami or bacon.  Or lots of barbecue. 

But for the cutlet type meats the fat issue tends to hold true.

Injection is a sort of speed brining.  

Koshering in salt can achieve similar results without the water.

Wet aging is faster and cheaper than dry aging and lets you sell moisture weight at meat prices though dry aging tastes 'better'.

First thing that comes to my mind; Let's say I have a dry aged piece of steak. Don't you dare cut that fat off! That fat is delicious!.  But when you're preparing chicken breast, you almost always trim the fat off. Chicken fat just isn't doesn't have that same appeal.  

Well, schmalz is a useful flavorful cooking fat. Dark meat remains more moist because it has more fat. And so I wouldn't say that chicken fat is undesirable.

It's more that fat in modern pork and poultry fat is not marbled through the meat as it is in beef. We also cook pork and poultry to higher finished temperatures in every cut than we do in the comparable cut from beef. So they often benefit from higher moisture added via brining.

In the curing and barbecue examples, those meats are cooked to or beyond 160 generally and so they too benefit from brining.

And one last thing. Beef and its fat is much more stable. It goes rancid more slowly so during aging, it doesn't add the off flavors you would get from poultry or pork. But even there, they benefit from a few days of letting rigor mortis pass before cooking.

Does the bacteria really break down the proteins or fat inside a dry aged steak?  I've always cut off the dark aged fatty part.  I leave just a little on.

It's enzymes (as opposed to bacteria) that do the breaking down of the protein during the dry-aging of beef. The process is called "proteolysis".

And I hate to ask this but what is happening to the fat as it ages?

Mostly it's getting oxidized.  But that's why you trim off that exterior part that was exposed to air.