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.
I'm adding this as answer because I really think this is a solution that will work for you, even though as written, I really can't call it a duplicate.
Check out this question and related answers: How can I turn off the power to an outlet if a water alarm is triggered?
The basic idea is to get a device that cuts power to the washer if water is detected. This will minimize any damage from a broken drain line since the machine will stop pumping water when the power is cut off.
Best Answer
One thing that is an option is automatic shut off valves for the washer that are linked to a water sensor.
Automatic Valves
Above is an example that I have experience with, but they come from several manufacturers and I think there are even "smart home" options that will alert you of a leak as well as turning off valves. I think I've also seen devices that cut power to the washer when there is a leak to stop it from draining water if the leak is actually a clogged drain rather than a burst hose.
With these valves, you still have to clean up some amount of water, but the problem is greatly reduced to the point of being able to clean up with a simple mop. Proper baseboard with a good caulking job will keep a small leak from getting behind the walls.