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You don't say what kind of cedar but they are typically too large for a container, however I included information on planting in containers. In the winter you can move the container to an unheated garage or shed and water occasionally.

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

This is a type of Deodar Cedar. They are only barely hardy down to zone 7, so you will need to offer quite a bit of winter protection until it is well established.

This article will help you with their care: https://www.gardeningknowhow.com/ornamental/trees/cedar/growing-deodar-cedar-trees.htm

Yes, they will need to be in the shade. this article will help:

https://www.gardeningknowhow.com/ornamental/flowers/coral-bells/coral-bells-plant.htm

It looks like very vigorous, new growth. This can appear discolored, but it will return to green.

Likely, this just got a little too much attention all at once. I would recommend giving the shrub slow, steady care throughout the year.

Because this is a Thuja, it will have the same care as its cousins, which are kept tall. This article will help:

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

Unfortunately, your photo did not come through. I am unable to identify the issue in question.

That appears to be Cryptomeria japonica, or Japanese Cedar. This article will help:

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