The lights on the dash always come on when a battery is flat. Once driven over 20MPH, They go out.
The wipers will not work when the bonnet is up. So once closed they will work.
The Fan will come on too when a battery is flat. Its happened to me, But after a little drive it switches off. Hence why the temp needle didn't move as the fan was on.
Regarding the engine struggling. That could be PCV.
Not too sure about the paint on the wing though.
I'd say go back, Have a little drive for around half hour and then see what you think.