You'll want some 'browns' to mix in with the clippings. Leaf season is coming around... pile up some leaves and run over them with your mower. Mix the shredded leaves in with the clippings as best you can. You can use any source of 'browns'. Browns are mostly carbon... wood chips, fallen leaves, hay, straw, etc... Greens, like fresh grass clippings, contain fairly high levels of nitrogen.The problem with trying to compost straight clippings is that they 'melt' into a stinky, messy blob lol. They will eventually compost on their own, but you'll better results by mixing them with the high carbon materials. I think the recommended ratio is 1 part green to 25-30 parts browns by volume. Btw... you can turn green grass clippings into 'browns' by letting it sit on the grass until it is dried out and brown. Much of the nitrogen will gas-off into the atmosphere. However, most people want the lawn to look nice, and if it clods-up it can kill the lawn. If you have too many clippings, then yes you can pile it up in an out of the way place, it will eventually compost on its own. But if you really want to put it to good use, then save an amount of clipping that will equate to a 'proper' ratio of your availability of browns as discussed above. The rest you can mulch back into the lawn. To mulch the clippings without special mowers or blades; simply mow the lawn and allow the clipping to mostly spread out, trying not to let it clump (perfection isn't necessary, but you may have to remove, or lift up the plastic cowl that you'll find on many mowers. Just put the cowl back in place near windows, people, pets, etc...) Let the clippings dry in the sun. Then mow back over them. They will easily shred and 'disappear' back into the lawn. This will both, feed your lawn, and your soil. It works best in the heat of summer, but unless it's a very thick lawn, you can usually spread it out enough to not kill the grass any time of year.

I think you have some very good materials there. I really don't think it have to be complicated science. Personally I would just put them down now, just mulch all your raised beds with them, I don't think it matters which goes first because it'll be composting over the winter and by spring time when it's warm enough for growing all the materials would be broken down. Problem with having a thin layer is that if you have just scattered stuff on top, they oxidized and dehydrates (if no rain) and doesn't compost well as in a big pile. It may be better to have grass clippings on top because it creates a good seal to keep the soil warm, retains the moisture and help the composting activities. If you were growing at the same time you put these materials down, I would put more broken down/composted materials down first, then the more green and coarse on top. Stuff like unbroken down woodchips can tie up the nitrogen in your soil a bit if you mixed in while it haven't composted. But since you mixed your wood shavings with chicken manure, and leaving it over the winter to compost, it is not a problem. I don't believe in doing all the complicated composting methods and separating things, organic matters is organic matters for me. So I would just throw them in. You can check out the youtube channel "OneYardRevolution" where the guy improved his soil simply by mulching with leaves and adding compost over time.

Referenced thread linked here. If you need to fertilize, just use a straight lawn fertilizer, no herbicides included. Many opt to go organic and use seed meals to fertilize or packaged organic fertilizers. And if you use a mulching mower, you can reduce your need for any fertilizer by 75% or more. A healthy, thick and sufficiently tall lawn will out-compete most weeds. And you can manually remove others if you stay on top of the game. Or if you feel the need, a spray lawn weed herbicide that you apply directly to individual weeds can be helpful. There are many good reasons NOT to use weed and feed products but the vast majority of them contain herbicides that have a pretty short half life and that breakdown rapidly when exposed to sunlight, water, soil biology and the composting process.

Glyphosate decomposes rapidly in the environment, so you should be able to compost that with no problems at all. Always good to avoid using treated grass clippings as mulch around live plants! Once it goes through the compost pile it will be fine.