I have three consecutive years of about 20 yards each composting in windrows in an open field that get turned regularly and we still get nasty annual and perennial grasses that pop up, even in the oldest stuff. Some people are also concerned with meds and supplements that were given to the horses, and we regularly find syringes w/o needles in the mix. Granted we get ours in a roll off dumpster, I'm sure if you are just loading a couple 5 g buckets you could pick them out.
It is good stuff, we top dress all veg and cut flower beds with it and see great results. Just have to be careful because pH can be all over the place and EC is variable. We also mix it in our large mixed compost piles during the growing season to offset all the green going in.
2,4d has a half life in soil of about two weeks, though it can be longer in aquatic situations. I believe once metabolized by the plant it is only a few days, meaning any weeds or grasses sprayed will be 'clean' faster than the surrounding soil. In both cases there is no remnant of 24d after a month or so. The hay has seed heads that either don't get digested when eaten or fall to the stall floor and get mucked out. Timothy grass is an usual suspect, as is johnson and barnyard grass, as well as seeds from cover crops.
Rabbit manure has the highest NPK, and milorganite (we all now where it comes from, right?) is 6-4-0 with some iron