2007 Prius- trouble turning on and off

I have a 2007 Prius with about 76,500 miles on it. I took it to the dealer for routine service at about 75,000 miles, and they told me that the 12v battery was very low, so they replaced it (I hadn't noticed any problems with the car before that). Since then, on two occasions (about a month apart), I have been unable to get the car to turn on first thing in the morning. I pushed the power button (with my foot on the brake as usual) and nothing happened. I tried this about five times with no luck. On both occasions, when I tried again after about fifteen minutes, the car started with no problems, except that the clock and pre-set radio stations had been re-set.

On several other occasions, when I have tried to turn the car off, it shuts off normally, but then when I open the door, it turns back on in "IG-ON" mode. I then have to push the power button again to get the car to turn completely off. When I start the car again after this has happened, the clock has often (but not always) been re-set.

Could both of these problems be related to the new battery (sorry if that's a stupid question...)? If not, any other ideas what it could be? I live about 80 miles from the nearest Toyota dealer, so if it might be something simple that even a car-dummy like myself could fix, that would be great!

Thanks!