There are epoxy products that can be used to repair stone, such as this one
This will work to glue the seams of the big chunks back in place. While it could be used to fill other breaks or gaps, it probably will not look very attractive.
Also, as DA01 says in his comment, the overhang makes this a very vulnerable area. Usually stones which overhang have a thicker dimension.
There is not too much you can do unless you want to put a facing under the overhang to help support it, a fairly big job involving cutting strips of matching stone. You could watch for the beginnings of cracks near the edges and fill them with an outdoor epoxy based material. This will reduce the chance that water gets in them and freezes, which leads to major cracking.
You could also tread lightly on the edges.
Obviously, the soil needs to be graded away from the house (in the back yard). If that means you need a retaining wall, then you need a retaining wall.
Regarding the swale, improved drainage sounds helpful. The real question is, why isn't water shedding (what is the real issue)? I've had swale issues, and the problem in those cases, was not with my swale, but the neighbors'... so water backed up in my yard. In such a circumstance, you can talk to the neighbors about the issue, ask them to fix it, offer to help, and/or sue them for damages.
But if the neighbors aren't to be blamed, then another alternative to keep the grass cut very short (grass slows drainage). Or you could check into a trench drain (concrete ditch). This is just another option that I didn't see mentioned, although it may not be your best option. As Tester101 pointed out, being there to see the issue is probably necessary.
Regarding power outages, gasoline or propane generators are the most reliable answer. A moderate generator should be enough for a sump pump, the fridge, and should keep the toilet flushing too. As you know, the drawback with a battery bank is that once it discharges, then you don't have a simple solution, like getting more gas.
Edit- Here's a possible way to use a retaining wall to increase the grade away from the house.
Best Answer
For my money, relative ease of installation, and the ability to remove and reinstall, I suggest using pressure treated or landscape timber, approximately 4"x"4" (smaller or larger to preference, budget and aesthetics).
Place these on the soil (not on top of the mulch), and secure them by driving a 12-16" piece of rebar through the timber and into the soil. Don't forget to pre drill the holes for the rebar.
Then some combination of soil and/or mulch can be piled/gathered on the hill side of the timber, to level the area to walk. It acts somewhere between stepping stones and a path depending on how much incline, and what the spacing is.
They can be serviced, removed or relocated as needed or wanted. The nice part is, if you decide to move the path, or plant different flowers, this is quite flexible.