I don't know much about the technical side of things but have been producing compost for over 50 years.
This is on open compost heaps and not in 'Daleks' and similar.
I put in virtually everything that I think would rot down or can add to the mix. A large proportion of my heaps (I have a lot of garden waste and therefore a lot of heaps) is made up of grass cuttings and leaves. Amongst this I add other green, non-woody, clipping and prunings (non-woody simply because I want to use the heaps fairly quickly), horse manure and ash from my bonfires (not too much ash).
I can't remember having seen any worms in my compost but I don't really look for them. As long as it rots down OK I don't worry about it. As I'm no longer fit enough to do too much heavy work I turn the heaps with a rotavator. This is much quicker and easier and gets done about once a month.
This area was emptied of compost last November and started to be filled with grass cuttings and leaves almost immediately afterwards and this is how it looked by the end of May. (If you're wondering why I have a large patch of nettles alongside it is because I leave it as a habitat for butterflies to breed, and at the end of the season they get cut down and put on the heap)
This was the last of that heap being turned, end of December, and then immediately spread onto the garden. The new batch of leaves for that heap are, to the right of the rotavator, will be spread into the bottom to start the new heap.
As these heaps are quite big they do generate heat but nothing particularly noticeable.
The horse manure is, of course, good nutrients for the heap and the ash (probably about a total of 4" deep on that heap) helps to keep a good consistency when spread through the heap. It also adds some potash.