Who is responsible for any incurred charges or fines?

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

Such charges would be your responsibility and you should discuss with your mechanic if they are driving your vehicle into an affected zone. 

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) your vehicle receives whilst in their care.

What happens if a mechanic asks me to pay the ULEZ / Low emission zone charge?

Such charges are the customer's responsibility - but the mechanic should first make the customer aware if they are driving the customer's vehicle into an affected zone.