Garage Floor Drain Alternatives
You can use a diamond blade to cut drainage channels. Noisy and messy, but it works. You can (for reasonable sums) either rent a concrete saw or purchase a diamond blade for an angle grinder, as you prefer. You can purchase a concrete saw, of course, but the sums appear unreasonable to me for a one-time project.

With such a limited amount of water (no wetness entering other than on the car) you could probably just cut a grid and leave it at that, with the water collecting in the channels and evaporating. An approach with a bit more finesse would cut the channels a bit deeper under the middle of where the car sits so the water would tend to drain away from where you are walking.

If you wanted to put in a sump (seems over the top for this garage) you would carefully slope the channels to it. While the floor does not have a proper slope, if the channels do, the water moves off the "islands" of floor between the channels, and then flows to the sump. That is typically done with a guide rail that you support at the desired slope above the floor to run the saw on.

I don't think the water in any way affects the structural integrity of your house.

If you have adequate headroom you could also add a new layer on top of the existing slab, without breaking it up, though I don't think there's any real need for that.

The cracks are probably not related and not of any real concern.