We are having trees trimmed and the branches put through a wood chipper. Can we use the resulting mulch to cover exposed roots on mature trees as soon as the come out of the chipper?
Yes, you can. Some types of green wood have slightly toxic effects to tender new growth, so only apply under mature trees. Make sure you don''t pile it up around the trunk; leave the root flare visible. Wood chips rob the soil of nitrogen while decomposing, so adding something with plenty of nitrogen will benefit the trees. Cottonseed meal, blood meal, fish meal, and hoof and horn meal (slow release) are good. Lawn clippings, like @Bob Jarvis notes, will work also, as well as many fresh manures. These do better when mixed with the chips before laying, or stirred in later.
If you renew the source of nitrogen now and then, to keep the chips actively decomposing, you're basically making a sheet of compost under the trees, which is very healthy.
Another thing to note: If any of the limbs you are chipping came from black walnut trees, do not apply them until you have found from a reliable source that the trees you are mulching tolerate juglone.
Make sure the mulch goes down no more than 3", or you may suffocate the tree. Any less than that is fine, also. And make sure you aren't piling it deeper in some areas. Go by the ground, not with level.
Yes, you can, though you've not said what variety of tree you're thinking of using it round, nor what trees are being chipped. Don't, though, put a layer more than 2-3 inches thick as a maximum, and if you have plants in that area too, they will need feeding in spring with nitrogenous fertilizer to compensate for the woodchips robbing nitrogen from the soil as they break down.
If there is a good quantity of leaves and small twigs in that mix that adds nitrogen content, you have Ramial Wood Chips which are a good nourishing mulch for fruit trees. Ramial Wood Chips are made from growth material (branches 2 3/4" dia. and smaller along with the leaves).
If it's mostly from larger branches, you'll need to watch the nitrogen uptake as the wood rots. It's all in the Carbon/Nitrogen ratio. The more wood, the more it will take up nitrogen in order to decompose.