I keep telling people not to run over the grass when they park. I'm thinking of getting some black edging barrier and placing it against the concrete, and then filling it up with soil to build the area back up.
That spot sounds like prime candidate for something like the Nitron product. It will work wonders. Something you can do without spending money for Nitron (although the Nitron will be great for the whole lawn if you don't normally aerate and don't want to do it mechanically. Also, it will help with germination and the new seedlings) is use a pitchfork. You can use a shovel if you don't have a pitchfork, the pitchfork works best. Drive it into the soil, and then rock it back and forth. You don't want to lift the soil from its bed, just loosen it. You might begin by driving it 3-4 inches and rock it to loosen that portion. Then, drive it deeper and rock to loosen the soil deeper.
LOL You went nuts on the weeds, huh? You'll likely get a crop of a bunch of others soon. These will be the warm season ones, like dandelions and clover. Leave them for the time being because you can't control weeds while trying to grow new grass. You can't use herbicides (if you might consider using herbicide) until the new grass grows up and has been mowed 3-4 times. You also want to stay off the new seedlings, so no walking to pull weeds for the first while.
Those purple-flowered weeds sound like creeping charlie (ground ivy) from your description. Creeping Charlie is a monster. Whatever you do to attempt control, you have to keep at it for probably 2 or 3 years, maybe more. The best time to do it is very late fall, around late October or early November. If you don't use chemical herbicides, then use garden vinegar (20%). Be careful with it though. It will kill anything you spray it on, so use on a calm day to prevent airborne spray. Two good ways to control the spray to direct to your target: 1) use some paper (several sheets or newspaper) or cardboard to shield the spray from desired vegetation, or 2) cut the bottom out of a paper/plastic cup or plastic soda pop bottle. Place the cup or bottle over the weeds you want to spray to isolate them within the cup/bottle, then spray the vinegar into the cup/bottle.
I am wondering if you think it would be a good idea to use the soil and grass seed mix in these areas.
I can't honestly advise you on that. I'm not familiar with the product, and I don't like quick fixes. Normally, products like that contain a ryegrass component, and that is often annual ryegrass for super fast germination, but it won't grow in again next year. When seeding, you want to do the whole area for uniformity so you don't have spots of different grass types. Different types are different colors (shades of green) and can look quite spotty when it grows in compared to the rest of area. Also, different grasses have different maintenance requirements, and you don't want to get stuck with one area needs mowing or needs water while the other areas do not. Whatever you use, use it over the whole lawn (the backyard I mean). The low spots can be filled with topsoil to level it off before sowing seeds.
As long as I rake and remove the larger rocks, is it ok to leave some there?
I think you need to remove it all because yes, it will interfere. Grass roots knit beneath the soil, especially Bluegrass because it also spreads across the soil. Beneath the soil are roots that grow vertically and knit and intertwine together. On the soil surface, or just barely beneath it, are rhizomes that grow and spread vertically. Roots and rhizomes need to be able to perform their functions without obstruction, especially so much obstruction.
You don't want anything in your garden bed or lawn bed that is not porous and does not decompose/degrade. An easy way to clear it is to use netting or closely knit/twined fencing to drag the area. It will remove the rocks and gravel, while sifting the soil. You'll have a mess, but that is easier to clean up and sweep the soil debris back into the bed with a broom than having to remove so much gravel and rocks by hand.
Because there is so much rock and gravel, maybe shovel dirt (soil, rock, gravel) onto the netting and then just shake, jog, jumble to sift the soil and carry the rest away. Sounds like a few trips but still easier than digging by hand. Maybe have a wheel barrel on hand.
No problem at all with the questions. Hope I helped a little. Tell me if I missed anything.
Just a side note, I think you should have the soil tested before planting the grass seeds.