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If this a new planting? New trees or shrubs will need plenty of water during it's first growing season. This article will help with that: https://www.gardeningknowhow.com/ornamental/trees/tgen/watering-newly-planted-tree.htm

There are many different types of Cedar trees. Browning of limbs and needles could be environmental or even caused by pests or disease. Here are some links with more information:

https://www.gardeningknowhow.com/ornamental/trees/cedar/growing-cedar-trees.htm

http://www.clemson.edu/extension/hgic/pests/plant_pests/shrubs/hgic2056.html

If it were significant damage, you would see death within a week or so. Usually they can handle some damage though.

It may change the overall shape of the tree but yes, it will recover. You may want to find out what kind of caterpillar it is that is eating the tree, and keep them off. I don't think it will do any serious damage to your tree. In fact, you might find that the tree grows back even more than before when it gets damaged. This is common as well.

Yes, this plant is considered deer resistant. Of course, no plant is completely immune to deer damage, but this one is a good bet for an area where deer are common.

The wood being stacked next to it won't pose much of an issue, but it may invite mold or other pest animals to stay near it. Other than this, there is really no issue in stacking wood next to it.

This, largely, depends on your growing zone, or climate conditions. These are relatively hardy when it comes to cold weather, and I would say you shouldn't have to wrap them. The spray is up to you, and can help give it a little more protection.

Here is an article that will help: https://www.gardeningknowhow.com/ornamental/trees/cedar/growing-cedar-trees.htm