Mike,
The months warranty is just a marketing thing, if you’ve bought from a dealer / garage you are covered under the UK Sale of Goods Act for Six Months.
Any car body shop can fix your paintwork.
Make sure you get the washer fluid sorted as if you get stopped by the police and they check you will get a fixed penalty notice, charged £60 and 3 penalty points on your license.
The dealer shouldn’t have let you drive the car with an empty washer fluid reservoir as it is against the law.
if the BMW Specialist has all the standard BMW computers / diagnostic equipment then you should find any gremlins lurking.
Don’t be surprised though if your car needs quite a bit spending on it due to the miles. Hopefully yours is newer with high mileage suggesting it’s all motorways miles cruising rather than start / stop.
All the best
Rich