Company wants to bring me onsite and gonna reimburse the plane tickets. Would it be indiscreet to buy them so I spend a week in the city?
No not at all, as long as you don't ask them to reimburse more than what they agreed to on rooms or flights. I also would not take advantage of it and fly home on peak time.
Ask. Say you’d like to spend extra time in the area. Nothing improper about that.
Just do a comparable to make sure you're not costing them more dough. Have that in your back pocket. "I have friends in the area and want to check out where the office is relative to them and where I could live". And you know what? Don't make it a surprise, just tell them that's what you plan on doing.
I've always stayed extra time to check out the city. Typically companies have no issue changing the departure date, but 99% of the time they won't provide any extra hotel nights or food. I've had a few pay for the rental car for the extra time though.
You could screenshot the return flight for both dates as evidence they cost the same.
A week yes. A couple of days to scope out where to live? Nah