Quackgrass is tough in an organic setup. Tilling would be detrimental. Plant cover and mulch will suppress it, but the rhizomes are strong and they will travel until they find a place where there's no mulch or plant cover. Be sure to mow your perimeter before it sets seed if you plan to smother your growing area with crops/mulch. If you can't mulch that huge of an area you may have to "hand pick" small seedlings until your plants are big enough to shade out the quackgrass. If you plan on using this land a lot you're going to want to plant a dense cover crop in times where you're not producing crops.

The bone and wood-material filled compost is a bit odd...and it may tie up some N, but if it's surface applied only a few inches your plants should be able to overcome that with their root system.

Sprouts will develop from the roots of the Cottonwood...remove as many/as much as you can.

It sounds like the owner of the land was trying to help, by taking down the cottonwood trees, and spreading mulch on the plot. Quackgrass can be pulled easily enough when the ground is damp, after a recent rainfall. Getting the tree roots out would require the use of a backhoe, to be accomplished in a day or so. I would think that you can garden on top of the roots, successfully. I can understand why you would want to rototill, but I think you can garden successfully with light hand tilling, and then planting in the top 4 inches of soil. We have heavy clay soil also, and I have spent 20 years making compost from tree leaves, and adding it to the garden every spring. The soil is easily worked now, but it took a long time to get there. I think you can have a successful garden despite the issues with roots and clay.

Quack Grass is tough in any environment and mulching only provides a better growing environment for it. Not even that very expensive plant poison, heavily advertised, glyphosate will keep it under control despite what the manufacturers tell you, and the researchers at Iowa State tell us that Quack Grass is developing resistence to that stuff.
That said, Quack Grass can be a very good green manure crop and can be suppressed for a short time by good mulching. I have killed off the top growth in new beds by laying newspaper, covered with something else,over the area but then I also had to go to work sifting the soil for the rhizomes and in clay that will be much more difficult than in my sand.
1. Unless that "compost" is tilled in it will not have an adverse affect on your soils available Nitrogen.
2. Tilling soil has been shown to be more detrimental than not tilling soil. If possible cover your planting area with newspaper and some material to cover the newspaper, and plant through that. The less you disturb your soil the better for your Soil Food Web.
4. The only "fix" for clay soils is to get adequate amounts of organic matter in that soil. Getting clay soils to drain can take some time but getting adequate levels of OM in will, eventually, do that. However, getting too much OM can result is soils that also do not drain well.
5. While one pest may indicate the presence of others I'd not get too concerned about that one.

I was thinking of planting through a type of black landscape fabric or the newspaper/cardboard mulch but won't the roots of the Quackgrass underneath compete with what I have planted ?
I was also thinking of just tarping the whole area to suppress the weeds and finding a cheap hoophouse or cloche and growing tomatoes & peppers in containers just so I can get some use out of the area while its being suppressed

Cheer up. You have to kill the grass, and you have to make sure those roots are dead. On top of this, you may have to add nitrogen (you may or may not). I do not see the situation as hopeless.

0) you have to border your lot, to minimize grass reentry. Any sort of deep enough underground barrier will work.

For the first two items, weed blocker for one year (or two), and hand pulling of suckers and weeds that make it through the plastic holes will do the trick. Black plastic will go well with a number of summer crops, but not with every crop. Certainly you can grow tomatoes, cucumbers, squash, peppers, eggplants, okra, melons. You can probably grow potatoes, chard, collard, and pole beans. The amendments you already have will provide nutrients and worm food.

Small seeded, cool season vegetables may have to wait until you are weed free and with a more crumbly top soil. Punch holes in the black plastic for your crops and weed whatever weeds come up through the holes.

For nitrogen, I suggest urea (not organic) or human urine if you can get enough (organic). In the Fall, you can rototill in a large amount of manure, and plant garlic in areas that appear to be grass free.

So: border lot, black plastic with holes carefully laid down, limited crops, watch for nitrogen, 2nd amendment in Fall. Next year you may need plastic in only part of your lot (I would still mulch with wood chips parts that will be without plastic). It is normal to take one or two years to get the soil the way you want it.