I recently joined Streetcar so I know the answer to my own question. You report damage to the car before you drive off using the inbuilt carphone linked to streetcar.
The Studley Court car doesn't seem to be available at the moment; it doesn't come up as an option for nearby cars. Don't know if it's permanently gone or just temporarily.
The nearest ones available are on Blackwall Way near the Ibis/Majestic, which seems brave of Streetcar, and there's one on the Boardwalk.
Let's hope the Studley one resurfaces.
By the way the FAQ on the Kengestion Charge is misleading. It says 'do I have to pay the Congestion Charge', and answers 'No'. What it means is that you don't pay it yourself; you'll be billed. Naughty.
They also misled me on the phone when I asked if you can reduce your excess to zero simply paying for one month, ie not monthly for a whole year which of course doesn't make sense if you only use it infrequently. They said Yes. Well, no you can't. If you don't take out the monthly payment for the whole year, you have to pay a daily charge of ?9.50 per day.