As to everything else: no, that would be creepy and the post office shouldn't do it. They don't need to become profitable or do off-mission things just to please the people that think it should be run like a business for some reason even though the whole point of the post office is to actually go everywhere (even where it wouldn't be profitable for FedEx et al to go) and be cheap (ie. not to be run like a business, to be run like a public service, which is what it is).
And I'm talking about truly hyperlocal data. When I load up Weather Underground or Purple Air, there are a bunch of sensors near me, but the closest is one mile away. I know the route of my postal carrier. He starts his day less than one mile and stays within one mile of my house his entire day. I'd love to have daily temp and air pollution readings from sensors that are near my house all day.
This weather data wouldn't be as useful as you think it would be. You'd be getting data only taken once per day more or less at the same time. You're much better using the airstrip a mile away that has a sensor that takes readings every 10 minutes or whatever.
And who cares if it’s not on mission? We can change the mission.
However, selling data to Google still seems like a bad idea. Things that are off mission and the point is to make money are just a distraction with no good purpose that's going to make the post office be greedy instead of just focusing on delivering the mail.
I could maybe agree that there was some value in having a public set of streetview like images (which is the more likely thing they'd do, not sell it to Google or whomever which would just be terrible), but it also seems deeply creepy. Then again, Google is already doing it so I suppose there's an argument that at least this way it's public domain.
And more importantly, that same data collected over time. Being able to monitor the daily changes in weather or traffic or erosion.
With the right contracts in place, it might even be profitable for them to go off route just to collect data.