For speed you really want to pile it up so that heat can build up. The thermophilic microbes are much more efficient than the cold acting microbes.

Yes you can layer the manure directly on top of the newspaper and then cover it with shredded leaves. I have constructed my raised beds that way and the tomato's loved it.I have also layered with manure and kitchen scraps .I mostly layer with manure,soil ,greens and any think that will rot.

I have read on this forum that horse manure has many more seeds that sprout than cow manure . Digestive process and all that . Don't be surprised if you get a good cover crop of various grasses from the grain feed in horse manure . All good for your compost pile . Go for it !

The objective of a Lasagna bed is not heat and the speed of decomposition since in many instances the bed is built and then planted in and heat would kill off most of the plants. All manures should be composted before going into your garden, because of the potential disease pathogens all manures have. Simply because our forefathers applied fresh manure, not knowing about those disease pathogens, to their gardens or fields does not mean we should continue following that practice, with that kind of logic you need not wash your hands.
Tree leaves have lots of nutrients they could contribute to your soil if put there instead of being thrown away as many people do.