Who is responsible for any incurred charges or fines?

Do I need to pay for congestion / ULEZ charges/toll charges?

Such charges would be the customer's responsibility, however, the mechanic should discuss with the customer in advance if they are driving the vehicle over a chargeable area. 

Do I need to pay for parking charges?

Mechanics are responsible for any parking charges for their own vehicles and any parking tickets (or any other fines) incurred whilst a customer's vehicle is in their care.