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This appears to be a Weeping Cedar Tree.
The variety or cultivator is undetermined.

Here is a link with care and growing information.
If you have not fertilized, the link below will help you.
This evergreen is quite low maintenance and has little issue with pests.

https://www.gardeningknowhow.com/ornamental/trees/cedar/growing-deodar-cedar-trees.htm
http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/fertilizing-evergreens-conifers/

This is Bagworms.
This article will help you.

https://www.gardeningknowhow.com/plant-problems/pests/insects/treatment-bagworms.htm

Absolutely not, it won't hurt the trees at all. Just make sure you don't layer the clippings too thick. Here's a good article about using grass clippings as mulch:

https://www.gardeningknowhow.com/garden-how-to/mulch/mulching-with-grass-clippings.htm

Root rot or a fungal disease are the most likely culprits. Overwatering or high rainfall could contribute to this, especially if water frequently gets on the foliage.

https://www.gardeningknowhow.com/plant-problems/disease/root-rot-in-garden-plants.htm

Keep the area around the cedar clean and free of fallen leaves. Once spring comes, prune off the blackened parts and dispose of them away from the garden.

Another possibility is sooty mold:

https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm

A slow-release fertilizer could help, but don't apply more than the recommended amount to avoid burning the tree. Evergreens generally need less fertilizer than deciduous trees, but they do need some:

http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/fertilizing-evergreens-conifers/

If it does start growing again to a similar size as it was when you bought it, it should be moved to a larger pot to allow it to grow larger.