Hi Garry,
I have a RaspiPi doing exactly what you are wanting to do. My RasPi sits in the Pool house, not the sprinkler box, but it is not climate controlled so it wouldn't be too much different (it's also in Phoenix and the temp in that room gets up to near 50C in the summer). The RasPi has been running unattended for quite some time - just did an uptime.
10:25:40 up 163 days, 2:25, 1 user, load average: 0.10, 0.17, 0.13
Can't remember why I rebooted it last time - I think I was moving things around and had to cut power to it.
HomeGenie is not running on that particular RasPi - it predates my HomeGenie install, but I installed HomeGenie on a separate RasPi and use http calls to control it. Once I get the HomeGenie configuration all sorted out, I'll probably consolidate the two.
The answer is yes to all of your other questions as well.
Good luck with your project,
Jerry