Well I'm sure Garyinchicago will chime in soon since he is practically a neighbor, but I will give my two cents. First step, get your soil tested professionally (if you haven't already done so). You say your pH is very low. This isn't conducive to having nice grass. You need to lime, but of course without a good soil test, you need to know which lime to use. It will probably be calcitic as most people are deficient in calcium, but you never know. So while you are waiting for your soil test you need to either kill the weeds, or else they will go to seed. If you don't want to kill the weeds because you will be left with just dirt, then mow them as you would a lawn, this will reduce the seed production. Applying a preemergent now will kill any weed seeds as they germinate. Use something that lasts 3 months so you can seed in late summer. Kill off your moss now with Moss Out liquid. It is an iron solution, non toxic, and very effective. The moss will turn black and decompose over time. Once you get your soil test back, you need to get all the nutrient levels up to snuff, and this includes raising your organic matter levels, so you might as well start applying organic fertilizers at a pretty high rate, once a month minimum. In July you will be killing everything and planting good grass seed. You can do this with Round-up, or Solarization, although solarization takes 2 months, and really needs a lot of sun to work well. So again, soil test now (www.loganlabs.com, $20 basic test) and mow or kill your weeds.

Maybe yellow nutsedge. Here is a link that might be useful: nutsedge

Pull it up now before it sets seed. That way you will have far less next year.

Whether you need to amend that soil or not depends on what is has now. How much organic matter is in that soil? How well does that soil drain? How well does that soil reatin moisture? What does that soil smell like? How is the workability (tilth) of that soil? What kind of life is in that soil? What is the soil pH? What is the nutrient balance? Start by contacting your state universities Cooperative Extension Service about having a good, reliable soil test done so you know the pH and major nutrient levels and then dig in with these simple soil tests to find out more about your soil, Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. 2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up. 3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart. 4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell. 5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy. Once you have this information at hand you will know what needs to be done to make your soil into a good, healthy soil that will grow strong and healthy plants.