Hello bobbyt
If the goal is to maximize nitrogen that you incorporate into your soil, the best option is incorporate it into the soil. Even a coarse turning-over will do. The reason: Nitrogen at the ground surface can be lost due to ammonia volatilization, especially in higher pH soils. For nitrogen present as nitrate, you can also lose N to the air, due to denitrification. [Edited to add this later: you won't lose the majority of the nitrogen, as it is bound up in the organic matter (as organic nitrogen), until it is slowly released by bacterial decomposition, and made plant-available. Nitrogen loss occurs from the "plant-available" forms of nitrogen, ammonia and nitrate. Some might be lost with runoff from the manure pile as well.]
If you are less concerned with nitrogen, and more concerned with just improving soil tilth, you can dig the manure in now, or next year. The benefit to digging in now would be to give the burrowing worms and other soil organisms more time to break down and re-distribute the manure in the soil profile. I would dig the straw in too (I assume the source of the manure + straw is well known, and there are no herbicide problems).
The black plastic, either directly on the soil or on top of your cardboard, might be helpful in controlling some weeds. My personal preference is to expose anything I am composting to as much air as possible to encourage aerobic decomposition.
Whichever way you go, your soil will improve with that manure. The choices are really related to what you want to accomplish with the manure, and how quickly.
Good luck with it!