Phd interview travel paid by interviewer - Is it bad form to ask to stay for the weekend?

If you make it completely clear at the time of asking that you would cover all additional expenses, then staying for a day or two extra would be regarded as perfectly normal and reasonable. (Academics travelling to conferences and workshops do this sort of thing all the time.)

Indeed, it may even be regarded as a positive sign: if you do join their PhD programme, then you'll be living in the city for several years, so the fact that you want to get to know the place in advance demonstrates that you're serious about your application.

My guess is that the somewhat intimidating wording "expected ... to depart on Friday" is intended to make it clear that you can't depart before Friday.

Just providing an extra perspective on this topic, but perhaps the "Arrive on Monday Leave on Friday" means that you are required to attend all 5 days of their scheduled interview process, and not necessarily that you are not allowed to spend extra time in their city. To me it seems they have that sentence in there to ensure someone doesn't arrive late, or leave early, or only attends 3 days of their workshop meanwhile the university is paying all travel expenses. Just another perspective to think about.

People in academia do this all the time. Just don't call it vacation or staying with friends, but rather say that you would like to stay for the weekend and get an impression of the city. Since it is a major concern most places that new Ph.D. students actually grows to like the place, such an initiative on your part will likely be appreciated.