Follow-up -- are my plans for fixing my driveway right?
If I understand you are suggesting that the grate go directly under the garage door, instead of in front. I had planned to make a concrete footer (almost a dam) that the door would close against, and have the grate in front. I was thinking that this would prevent the door from getting in contact with the water, freezing closed etc - problems I have now. But then, with the grate under it, it should drain anyway.
I had meant that the full width grate be in front (outside)of the door, stopping water from reaching the door. I'm not a big fan of a dam as it will trap snow, melting off a car, from draining out of the garage (unless you also have a floor drain in the middle of the garage). I suppose that having a grate below the door will work, but it will allow wind and perhaps wind blown snow into the garage.