Yes, but a better challenge is to go from Wrest Point via Battery point to Salamanca Place and return. Plenty of uphill and downhill and plenty of places to refresh. I have done it four times and I am in my eighties.
As we did only one night I am unable to help, however the area doesn't seem too busy.
Yes you can run along sandy Bay road
I'm not a jogger, but you could, but I would think smaller side roads or a long the water would be more pleasant. Sandy Bay Road is a main thoroughfare and the sidewalks weren't very wide. I'm sure the hotel staff could direct you to better streets.
Yes, plus there is a dedicated bike track.
As Kristine said, yes, both sides, right into town
Get an answer from Wrest Point staff and past guests.