I have a friend who regularly uses several inches of chicken poop and bedding in his garden. He does not get burning, but he does get very enthusiastic green growth. This can sometimes be a problem. For instance, he had beets with tops as tall as a person. His tomotoes, peppers and squash did just fine with all the manure. His soil has become incredibly mellow over time, and you can easily sink your arm in to the soft material (preferably before the manure is added!). I would just go ahead and put it on the garden.
Renais

If you have raw organic matter in that mix you do not want to till it into the soil because then the soil bacteria will get busy and digest that raw organic matter using large volumes of Nitrogen while doing that, possibly to the growth detriment of your plants. Just spread that on as a mulch.
The time to apply lime to correct soil pH is about 6 months before you want that soils pH to be corrected because of the time it takes for the reaction to take place in the soil. There is an old, and mistaken, thought that applying lime and Nitrogen together will cause the N to gas off, be lost to the atmosphere, too quickly. So could you apply your lime and your chicken house litter at about the same time? Yes, although it is not the best thing to do since the lime will have too little affect on the soils pH this growing season.

(I could be wrong about this, but I think dolomitic lime works immediately. I'll go read up.)

I use my deep chicken litter on my gardens in the spring---fresh from the house. My soil can handle the nitrogen.

Yes, you don't have to bury the uncomposted leaves. I do lightly fork my litter in before planting a row.

And you probably don't have to worry about the nitrogen borrowing that kimmsr mentions, because your litter is combined with a high nitrogen material---the chicken manure. You won't see the effects of nitrogen robbing, although you might see the effects of too much nitrogen.

If you are worried about your nitrogen levels, put that June litter aside. If you aren't, you can use it as a mulch, although I'd be more inclined to use it as a side dressing.

thanks for the help folks,

i must show my ignorance now, what is side dressing?

It makes no difference if you use calcitic or dolomitic limestone, what would make some difference would be if you used hydrated lime, but it will still take some time for the reaction in your soil to cahnge the pH.
Side dressing is putting some of your material down along side the plants that are growing, most common in row cropping where the "fertilizer" is poured in the row between the plants.