It’s good that you recognize grass has different requirements than garden plants. Since your soil is most likely acidic and coffee grounds are also acidic, that would be moving you in the wrong direction. Dumping stuff willy-nilly into the soil sounds like a terrible plan to me. Personally, I use the coffee grounds as mulch around the blueberry bushes and put the crumbled egg shells out by the bird feeders.
You should be aiming for a pH of about 6.5 for the lawn. Lime and fertilizer shouldn’t be applied at the same time (at least a week apart); it would be fine to put the lime down now. Weeds like acid soil. Getting the pH right is a natural weed preventative.
Getting the grass seed off to a good start will also go a long way toward preventing weeds so I strongly suggest you plan on applying something labeled as Starter Fertilizer when you plant. Do it just this once. A dense, healthy patch of grass keeps weeds out all by itself; grass seeds like to germinate in the dark but dandelion seeds need light to germinate – that’s why they pop up in bare spots where the grass is thin. Covering the area with straw after you plant will also help prevent the weed seeds you’ve tilled to the surface from sprouting.
As for tilling in soil amendments other than lime, that sounds an awful lot like work to me. I prefer the “better living through chemistry” route so I can’t help you there. Me, I wouldn’t even till the soil; I’d use a general herbicide to kill existing plants and then just scratch up 1/4 inch of the surface.
Here’s my plan for you, with a few optional steps:
Test the pH.
Apply sufficient lime to reach 6.5 (but no more than 100 pounds per 1000 square feet. If you will need more than that, apply half now and the other half in the spring.)
Till the soil and level. (I would use a RoundUp type chemical to kill the existing stuff first. Your call.)
Spread the seed and very lightly rake in. (I would also use a light roller to insure good contact of seed to soil.)
Spread Starter fertilizer.
Cover with straw.
Water according to instructions on the bag of seed.
Enjoy your new lawn. Throw down some of whatever kind of organic fertilizer suits your fancy each fall and check the pH every couple of years.