Can I do this body work myself?

It depends on several factors. Like if you want it perfect - get a professional to do it, unless you've had experience before - or are prepared to have it off the road for long enough to do it properly. Is it plain white - or blizzard pearl? If it's blizzard, forget even thinking about it for your first spray job.

Spray painting - I practised on cardboard boxes and other pieces of steel that I could lay my hands on to get the technique right, and ended up with a good job - but it was slow work, and several panels I had a couple of goes at. That was on a car which was off the road and I was restoring.

That was back 20+ years ago when you could spraypaint in your backyard - these days, in many municipalities you can only do it in a booth with proper extraction. There might be spray-paint booths you can hire, but they're not cheap. Plus cost of a spray gun, compressor, respirators, paper, masking etc needs to be considered. Though - thinking about it, it's probably water-dispersed paint, and I understand it's a bit more complex.

The hood/bonnet is aluminium - not sure how you'd approach that - I've tapped out steel panels, filled and painted, but aluminium is quite different I believe. Early steel was easy to beat (compared with more modern steel). The rear looks like it is probably just needing painting - unless the alignment is compromised - is there any hidden damage to the structure behind. You'd have to be sure that there is not damage to crush boxes, mounts or reinforcing bars. And you'd have to buy a tail-light by the looks of it.

 

Many times insurance lowball the estimate. Professionals know how to work the system and provide proper evidence and documentation to receive more funding. That work looks like it should be more than what you received. For minor scratches, there are some ways to repair with primer base coat and clear coat, but for your color which looks like blizzard pearl like my car, color may not blend perfectly and it may very noticeable. Sadly, you pay for what you get.

 

It might be worth checking if one of those mobile repairers can do it for a lot less $ than a body shop. They can probably tease out the hood dents with their special tools if there's no paint damage and they can get access to the back of them. They would also be better at matching the paint on the rear corner damage if they can do it.