I still think you would be better off xeriscaping:
https://www.daviddarling.info/images/xeriscaping.jpg
https://www.google.com/imgres?imgurl=htt ... CC4QrQMwAQ
https://yardfairy.com/wp-content/uploads ... CN1770.jpg
water is a precious resource...
But if you insist on having lawn and since no one else has answered. ... You can't put down grass seed with all the weeds there; have to get rid of all the weeds first. You can rent a little tiller and have your small plot tilled up in an hour or two. Then water it and wait a couple weeks, let all the weed seeds you just brought to the surface sprout. Add a bunch of bags of enriched topsoil, compost, well aged composted manure, etc and till it all in, turning the new weed seedlings under in the process. Sow your seeds. Cover very lightly with soil (something very light like potting soil would be good) and press it down so seed is in contact with soil. Water gently every day until grass is well sprouted and every other day for awhile after that while it is putting down roots.
But you understand in the middle of summer is exactly the wrong time to do this. You have lived with it dead this long, live with it until the beginning of Sept. and then do the steps above.
Your shady areas could be dug and planted now. Do steps above, except just turn it with a shovel. Plant with shade plants, which could include:
bellflower, heuchera (coralbells), spiderwort, star flowered Solomon seal, western bracken fern, lady's mantle, maybe a dwarf hydrangea, snakeroot, pulmonaria, silver lamium
look for a good nursery (NOT big box) especially one specializing in native plants, which will be hardier and more adapted to your situation, so lower maintenance. But understand anything you plant will need good care the first season while it is getting established.