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I would certainly trim away the dead plant material on this Arborvitae.
You could blend in the narrower bottom portion of the plant with the fuller top part to give a better appearance.
Mulching the base and fertilizing with an all purpose fertilizer would be beneficial.

Here is a link that has more information.

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

How about:
Acer palmatum
Snowy mespilus
Pink berried rowan
Pyrus calleryana
Chinese Paperbark Maple
Siberian Pea Tree
Judas Tree
Myrtle
Chilean Fire Bush
Tree Peony
Plumosa Aurea-Sambucus

The image did not show up. The picture would more than likely help me to identify the problem. It would be very hard to say exactly what caused it. This article will give you an idea of what causes browning like this for general evergreens: https://www.gardeningknowhow.com/ornamental/trees/pine/pine-tree-drying-bottom.htm

Yes, that is more than likely a soil infection. I noticed it is pretty wet as well. If it remains like this for too long, then this can happen.

Just apply wettable sulfur and about a tablespoon of dolomitic lime. It should clear the infection in the soil. It won't make the dead growth turn green, but any new growth will be healthy. You can cut out these dead spots if you wish.

Conifers have shallow, extensive roots. You will be cutting into roots in order to plant which isn't the best for the tree. In addition, the tree will consume most of the moisture in the soil making it nearly impossible for other plants to thrive. I don't think an azalea stands a chance. Do an internet search for perennials/shrubs/annuals for dry shade in your region. I would start with a very small plant, either a plug or something you grow from seed. It's going to be difficult, but possible, for the new plant's roots to find spaces where moisture remains A small plant will need less water during establishment than a larger plant so is more likely to survive transplanting. Plus, as mentioned, you will be destroying feeder roots of the trees to make a planting hole so you want to minimize the risk to the trees. Anything you plant will soon be infiltrated by tree roots. Look for plants that grow deep roots, like many native plants, or ones that have taproots. Alternatives are garden statuary, fountain, fairly garden or growing in containers.

The short answer is, yes, but if you are very familiar with keeping bonzai trees then this can be done. The roots will require constant maintenance, and trimming... But very properly. These trees are about 30 feet tall in nature, so keeping them 1/30th of that height will be very difficult. They also don't trim as well as their close relative, the Leyland Cypress. I would suggest crawling junipers. They will be more tolerant of confined spaces, and often won't exceed 2 feet in height.