Yikes! That is serious indeed. You say you mentioned the issue to the builder - was mitigation of the issue placed in your purchase contract?

If not, and 14 months has elapsed, my gut feeling is that you are SOL.

Edited to add: getting those cars out of your garage would worry me sick . . .

For sure don't drive on the unsupported portion.

It's preferable that drainage be solved at the ground surface by proper grading, if at all possible. Then, one does not need to worry about underground pipes clogging or their capacity being maxed out. It looks like there's a lot water being handled. (So much for the ability of loose rocks to counter erosion!) If I understand what you're saying about the vast bulk of water entering at the mouth of the drive, we see it commonly solved by people adding a speed-bump-like hump at the drive entrance where it joins the street. Unfortunately, it's a band-aid. It works but doesn't look all the great. A slightly upgraded band-aid would be to cut out a cross strip of drive and reshape a hump into it. It would be good for you to add a picture taken from the street showing the entire path up to the garage.

On another point, if this was my situation, I would want a 6" pipe attached to the slot drain, not a 4" pipe. I think the 4" pipe is too easily overwhelmed in a major rain.

This is a defect in how the builder did the final grading on the lot. I'd contact them and honestly, at least have conversation #1 with a lawyer to see what they think (because most builders will tell you to pound sand, which you can't because IT ALL WASHED DOWNHILL).

How the hell are you going to get your car out?