Dealing with water on a bad garage slope
I have a similar situation, though due to poor masonry work rather than settling. Meltwater runs to the side of the garage near the house entry door, soaking the stair base and a shelving unit, along with any objects sitting there, with saltwater. 

I intend to cut several channel drains into the slab at key locations using a rented diamond saw. Normally this type of drain would connect to a pipe running to the exterior (a "daylight" drain), or would have a deep drain pail below. For our purposes, this probably isn't necessary. I may excavate a small area below the drain unit and add pea gravel to increase capacity, though. 

The idea is to catch the bulk of the water flow as it leaves the parking area. Water will settle into the soil below the slab. I don't believe that it'll be enough to cause further settling or other issues. We're talking about a few gallons per week at the most, and only during winter.