I'm a PT. Will try to answer your questions from my perspective...
How often do you see them? How long are your sessions?
For me, this tends to be client-led. I see some clients twice a week, some once a fortnight. Once a week is typical.
How hard do they work you? Proportion of strength to cardio?
This varies depending on client goals. Sometimes it isn't just about working 'hard'. If I'm establishing a fitness baseline I'd expect to push a client to their limit, but it might take some experimentation to work out where that limit is. And of course it will vary according to equipment; some people will be able to row or cycle all day long, but can't do more than three push-ups or will be out of puff within two minutes on a treadmill.
The first session with any client is always a bit of a compromise because I'm taking a history, working out goals, talking about likes and dislikes, covering essential health and safety stuff, measuring height/weight/HR/BP etc. It's usually session two when clients walk out wobbling!
Do they advise you on workouts to do without them in between sessions?
I do this, some PTs won't. Again, it rather depends on client goals. I design programmes for my clients that they can do in their own gym time.
How much do they advise you on diet?
I don't advise on diet. Unless a PT has an additional qualification over and above the PT diploma, they shouldn't be advising on diet beyond pointing you towards the standard healthy eating guidelines. If you want tailored diet advice, you should see a dietician.
How much do you pay (if you don't mind saying)? I am outer London so expect to be higher than elsewhere in the country
I charge between £60 and £75 an hour but I sell packages, not individual sessions, so it depends on the package.