Wait until mid August to start this project. It is too hot now. Fescue is the only grass choice for that much shade. It is not very forgiving of dog digging or wear and tear so just know that in advance. Get the project going in August to give yourself time to add more seed if you see thin spots. 1. Fertilize the weeds so they will die quickly 2. Kill the weeds with Round Up which will kill anything it gets on 3. Water the area every day as if you were trying to sprout seeds 4. After a week of daily watering, spray the new weeds with Round Up. 5. Wait a few days, rake up any dead foliage, and seed. Roll the seed down with a water roller to make good contact with the soil. 6. Water 3x per day for 5 minutes to keep the seed moist. Fescue seed should spout in 2 weeks. 7. Mow when the new grass is 4-5 inches tall. Mow it down to 3.5 to 4 inches. 8. After you have mowed for the second time, you can apply a synthetic fertilizer. Or you can apply organic fertilizer any time. After you have mowed the grass once you can start to back off on the frequency of watering and increase the duration. By that time of year your goal will be to water 1 inch of water, all at one time, every 2-3 weeks. It might take some time to get there but that is the goal. Next year you will gradually move from monthly watering up to weekly watering as the temps increase.

It's unlikely that you have warm season grass. The reason it doesn't go dormant in the heat is that you're watering it enough to keep it green. The most common lawn grass for the Wasatch front is Kentucky bluegrass. A lot of "drought tolerant" mixes, especially cheap ones, have a high percentage of K-31 (sometimes called KY-31 or Kentucky-31). Stay away from these. Despite the fact that the name has Kentucky in it, it's not Kentucky bluegrass. It's a pasture form of tall fescue. It's very hardy, but it's wide bladed, grows faster than other lawn grasses and tends to get clumpy over time. Picking a time to seed is tough. You need to wait until the heat of the summer breaks and that may not be until some time in September. Then, you want to make sure you plant 6 weeks before the expected first frost, which is early October. Oops, only 4 weeks after heat broke. In the spring, if you wait for all the snow to melt, you'll almost certainly be planting too close to summer heat, so the grass won't have a chance to establish before the heat kicks in. I've had the best results with dormant seeding. To dormant seed, wait until just before the first good snow and seed at a slightly higher than normal rate. The normal freeze/thaw cycles and any snow melts will carry the snow into the soil. In the spring, the melting snow and occasional rain will provide enough moisture for the grass to germinate.

Looks like dallisgrass to me. If you can post a picture of seed head from it that would be a dead giveaway.

Poa T (rough bluegrass) possibly?