How much of a factor is shade in your yard?
Do you know what kind of grass you have (fescue, rye, and Kentucky bluegrass being the three likely choices)?

Be very careful about bringing in more soil, More soil always changes your drainage. Mulch and compost do not change your drainage, so you can use those freely.

You might want to remove the rock and carefully replace that volume of rock with an equal volume of soil and/or sand. Tamp it down with your feet and the full weight of your body until it is even with the surrounding soil.

Can't help much with the grass issues without knowing about the shade and what kind of grass you have. I suspect you have KBG in the shade and that is source of all your issues.

Generally spring is a poor time to be seeding new grass because the summer weeds are germinating at this time, too. It is also poor because the young grass will not have developed strong roots which can survive the summer heat. In Seattle that might be a different story with the heat. You can likely water your way through the summer with little loss of grass.

Here is the beginner's guide to lawn care...

Basics of Lawn Care

After reading numerous books and magazines on lawn care, caring for lawns at seven houses in my life, and reading numerous forums where real people write in to discuss their successes and failures, I have decided to side with the real people and dispense with the book and magazine authors. I don't know what star their planet rotates around but it's not mine. With that in mind, here is the collected wisdom of the Internet savvy homeowners and lawn care professionals summarized in a few words. If you follow the advice here you will have conquered at least 50% of all lawn problems. Once you have these three elements mastered, then you can worry about weeds (if you have any), dog spots, and striping your lawn. But if you are not doing these three things, they will be the first three things suggested for you to correct.

Watering
Water deeply and infrequently. Deeply means an inch (as measured by tuna or cat food cans placed around the yard) in every zone, all at once. Infrequently means monthly during the cool months and no more than weekly during the hottest part of summer. Do not spread this out and water for 10 minutes every day. If your grass looks dry before the month/week is up, water longer next time. If that does not work, then you might have to water every 5 days during the summer's hottest period. Deep watering grows deep, drought resistant roots. Infrequent watering allows the top layer of soil to dry completely which kills off many shallow rooted weeds.

You will have to learn to judge when to water your own lawn. If you live in Las Vegas your watering will be different than if you live in New Orleans. Adjust your watering to your type of grass, humidity, wind, and soil type. It is worth noting that this technique is used successfully by professionals in Phoenix, so…just sayin.' The other factors make a difference. If you normally water 1 inch per week and you get 1/2 inch of rain, then adjust and water only 1/2 inch that week.

Mowing
Every week mulch mow at the highest setting on your mower. Most grasses are the most dense when mowed tall. However, bermuda, centipede, and bent grasses will become the most dense when they are mowed at the lowest setting on your mower. In fact there are special mowers that can mow these grasses down to 1/16 inch. Dense grass shades out weeds, keeps the soil cooler, and uses less water than thin grass. Tall grass can feed the deep roots you developed in #1 above. Tall grass does not grow faster than short grass nor does it look shaggy sooner. Once all your grass is at the same height, tall grass just looks plush.

Fertilizing
Fertilize regularly. I fertilize 5 times per year using organic fertilizer. Which fertilizer you use is much less important than numbers 1 and 2 above. Follow the directions on the bag and do not overdo it. Too little is better than too much. At this point you do not have to worry about weed and feed products - remember at this point you are just trying to grow grass, not perfect it. Besides once you are doing these three things correctly, your weed problems should go away without herbicide.

Oh and about that root...

That particular root is not taking up any water at all - None! All the feeder roots are 15 feet away and slightly deeper in the soil. That root has become one of the structural roots holding the tree in place. If you want to build up that area with soil, that is fine, and then seed with grass over top. As the root increases in diameter, it will poke out again over time.

Thanks.
I had my gardener dethatch, reseed and spread peat moss this week. I think you mentioned this is too early. I thought I was late.

I am also surprised by your suggestion of watering only once a week even in the hot summer. I water 10-15 minutes everyday.

Am I reading this correct?

Are you suggesting that I water only once in March and April?