Might be easier to sort the scratches on the hatchback if you remove the Jazz badge first.
It's only stuck on with badge tape, which you can cut through with fishing line, although I've heard some people use dental floss!
You can clean off any residue with white spirit or Meths and re-apply the badge using new tape when you're finished.
As for the front end, it won't just clean up with elbow grease. Looks like a respray is required as the paint has delaminated.
Someone like Chips-away might be your best course of action - for both ends of the car.