First, as it applies to all points:
yes, the temperature it will start in is unknown, but at the same time there won't be a lot of cold starts. This isn't a commuter car, it's an emergency generator. I assume there's no automatic start or transfer switch. If you decide to fire it up, it will probably be on for a while. Does the power go out a lot where you are? As long as the oil isn't so thick that it interferes with starting, I wouldn't sweat the wear too much.
1) it might be nice to have something designed with storage in mind. Like Amsoil ASE. Supposedly lots of anti-corrosion agents in there for OPE that is stored.
2) I see cold temperatures as a non-issue. You are *RE*-filling. The engine is warm. Run anything you like. The bigger concern is...
3) Now, extra-hot temperatures is another matter entirely. I may be concerned below 101F. Do you load your genset to 100% or does it lope along at 50%. If you're not overloading it, this may be less of an issue as the designed air cooling may be sufficient. Is the genset in the sun or shade? Do you have to account for the beating sun in the heat the engine needs to shed?
4) See #2. The engine is warm, use whatever gives you good runtime and protection. I presume that when your "outage" is confirmed to be over you'll change the oil again for storage from the way you've written in this post. I'd be concerned about selecting a re-fill oil that gets you the longest runtime between changes, since it's running continuously. Perhaps you could get 2, 3 or 4 days per OCI instead of just 1. This might be where HD40s or 15W50s come into play?
5) Amsoil ASE has lots of ZDDP, and is a good storage oil (although maybe pricey to simply throw away once a year). Perhaps move it from your genset to your lawnmower or something so it at least gets used. Any moisture will simply boil off as long as it isn't rediculous in quantity.
Of course if you think too much about all of your different scenarios, you'll have to have a ton of different oils on hand. You certainly don't want to depend on going out and buying something you need (including fuel) when the power is out. Remember Aug/2003? GC or Amsoil ASE might be all you need. A mid-winter start with ASE might be a little tougher than desired, but would it wreck the genset? It's still a 10W.
If you could work out other equipment to use the GC and ASE you could do seasonal OCIs on the genset. GC for winter standby (and consumption in something like a road vehicle the following summer if it was never used) and ASE for summer (with consumption the following winter in something like a snowthrower if it wasn't used?)... Just trying to avoid waste of good oil