On weeds, my cows loved butter cups and some other selective weeds.
I used Grazon P+D one year. The P was for brush and the D was 2-4-D, a popular broad leaf herbicide. I don't have any brush any longer and quit renewing my applicator license so I can't buy it. But I do go to the local farm store and buy D in quart bottles (not restricted) and mix at 1% (1 gallon in my 100 gal tank). D is a broadleaf herbicide and that's what you want for weeds in a grass patch.
I like D as it works the year you put it out and keeps working. On completely eliminating weeds, unless you spray often forget it. I have a field I have worked with a crop for over 30 years and every spring here they come. I notice it mostly when I have plowed the field. I guess plowing brings them up to the top where they can germinate.
On getting cleaner cuts later on in the year, a lot of people around here will want the second or third cutting for their hay as it usually has fewer weeds.
On fertilizing with chemicals, it depends on your soil and what was grown there as to what you need and what the plant you are growing needs. As stated by others, you should have an Ag. and Mech. (A&M) University somewhere around where you can send soil samples, tell them what you want to grow and how much and they will come back with what you have in your soil and what you will need to supplement it to grow what you said you wanted and how much of it. Ours has a www and a form to fill out, how to take the sample and how to ship. Real handy. Costs $10 per sample and they send you an email back in about a week with the answers. On my samples I go around the field and take several samples, mixing them together to get an averaging and send that in as one sample for $10.
I like to put my chem out about a week or so before I plant to give the plant a chance at getting it before the weeds take over. On grass, the popular grass here is Coastal Bermuda which is a long stemmed semi tropical grass, makes great horse hay (picky owners) which wouldn't be good for you but you can get on the www and find grasses that are perennial and suited for your area and your intended usage. For cows I plant annuals like the mess I am going through this year. Can't believe too good of a crop is a pain but this time it is.
Anyway, that grass is a warm season grass so I wait on the fert. till the temp gets up before fertilizing. I go out in the field and look for the new grass shoots for that season, telling me that the plants have started growing. Then fert.
Some N fert is urea and requires special attention to get it in the ground before it evaporates. Others are Ammonium Nitrate and Ammonium Sulfate. The sulfate is sulphur and some soils need that. It also is less % per wt but is less apt to give you grazing problems like the Nit. Too much Nit under adverse growing conditions could produce Nitrate Poisoning in cattle. Both are good to be absorbed into the soil without evaporating when broadcast like you would do over a grass field. Always best to fert right before a rain to prevent any evap loss but who can always be that lucky.
Oh, and chicken poop is the hottest of the available poops by a long shot.
These are a few things I learned from the school of hard knocks over the years.
HTH,
Mark
