You cannot do a basic level with a machine on wet soil. That will compact it more. You need to loosen the compaction first. This is true compaction done by a machine on wet soil. The weight of the machine squeezes air out of the soil which is replaced by water which then evaporates. Your soil will be hard as concrete when it dries. You should not need more topsoil unless you have low spots in the yard that don't drain well. Bobcats do not do a great job of surfacing the soil with a traditional box blade, but your yard is rather tight with trees and shrubs. I would be tempted to use something like a Harley rake as pictured here. You undoubtedly have a different name for the tool in NZ. These machines work only at the surface of the soil and can be adjusted with much more precision than, say, a hand held rototiller. There is also a rototiller attachment that fits a Bobcat that would go a little deeper into the compacted area. These are much better controlled when on the Bobcat. The problem with hand held rototillers is they buck up and dig down depending on the soil softness, rocks, roots, and pipes. This up and down action leaves an uneven subsurface and an unequal depth of fluffy soil on top. When that settles it settles down to match the uneven profile of the subsurface. But when you use a machine mounted rototiller, you don't get that. The result is much better. This should help you talk to the Bobcat man. Then you can level it by hand after the surface is loosened. There should be good drainage out away from the house. Bringing in more topsoil usually interferes with drainage, so don't plan for that.

It looks like they poured a new slab on top of the old one, it looks like you have some moisture issues there also, this is why they probably poured a new slab over the original rat slab and they just didn't/couldn't get under the stair to level it so they just pushed it under there and left it. You can get a quick patch cement and fill the hole. How old is the house?

Here are 3 of several responsibilities of a General Contractor: Organizing and managing the jobsite. Hiring subcontractors and managing their quality. Contracting suppliers and vendors. "so I went ahead and hired a guy to install the vinyl planks." Why?"

How old is this house. If built in 1900-1920 the wood milled then was truer to size. That rot didn’t happen recently. It’s been building up over time. You probably need yo replace both window and framing to properly fit new window to eliminate problem. I would be concerned about wood condition below this window too. Water damage might go further. Best to figure that out too. Also, get checked for termites and carpenter ants. I know, it’s a pain but that’s just a part of home ownership. Good luck!