Weather: could be anything. The tours won't be especially weather dependent anyway. I haven't been to Kentuck Knob, but the standard Fallingwater tour doesn't go outside much. You could see unseasonably warm in early November, or it could be snowing. Hard to tell. If normal, expect temps in the 40s to maybe 50s, and hopefully it won't rain. (Sun? Ha!. We have more overcast days than Seattle I think.) Probably a bit late for fall color, though there may still be some.
It'll be a good idea to make a reservation, even though it's a weekday in November. If you go to the trouble of getting down there with a hired car, you want to make sure you can do them both. Getting between them you probably only need, oh, 1/2 hour I think, so you could schedule the time of whichever one is second (doesn't matter which order you do them in) about an hour after the first tour is meant to end. Add more time if you're paranoid there, but you may be paying hourly for your car so consider that.
Getting there: this is a tough one. Obviously it would be a heckuva lot cheaper if you did drive. You're talking one tank of gas and one day's rental at probably a pretty inexpensive rate, 20-30 bucks for midsize. The "limo" companies are probably about it. I don't think there are any dedicated car service companies in town. Many of the limo companies do have sedan service, though. One company that had its prices online says their Town Car is $298 for 5 hours and $47 per additional hour with no extra charges, even includes gratuity. (Edit: I noticed as I was about to close that window that it says "Prom" prices. I dunno if your use could be negotiated lower. Perhaps.) That's going to come out to a lot more than you're getting at $350 for 15 hours, which seems like quite the bargain to me. I don't have much experience with these companies, but if you want some names and numbers feel free to PM me. Maybe someone else from the area has a recommendation.
Beyond that, though, I wouldn't be too confident that many of these drivers have a clue how to get there or get around down there. I'm sure there have been a few people over the years who hired a car to drive them from here to Fallingwater, but finding a driver who is actually familiar with getting there I suspect will be unlikely. I wouldn't feel too confident leaving the navigation to any of them, even though if you tell them where you're going I suspect they will be gathering map info and stuff.
If you're absolutely positively not going to drive, then finding a FT'er or friend of an FT'er could help you keep the cost down. I can't volunteer for this, sorry, nor do I know anyone directly that I could recommend. Same caveats apply for knowing how to get there. Even someone who has been there once is suspect. Pittsburghers are notorious for only knowing how to get around very small areas near their home and to and from work.
Curious what the aversion to driving is. It's not a particularly complicated routing, though some of it will depend upon where in the Pittsburgh area you'll be staying (if you even know that yet). From the PIT area or even from downtown it's a fairly straightforward drive, and driving yourself would be by far the lowest cost and most flexible.
Kinda surprised, really, that there are no one-day bus tours. Seems like a logical thing to offer, although it may be an odd season for it. Probably not too popular on a weekday in November.
Anyway, let me know what else I can do to help.