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It sounds like you have done about all you can thus far. Bt (Bacillus thuringiensis) products sprayed in the spring can be successful in the control of bagworms. In addition to hand pulling, power washing the 'bags' off the plant may be an adequate form of control. Here is an article that you may find helpful: https://www.gardeningknowhow.com/plant-problems/pests/insects/treatment-bagworms.htm

Fall is likely the best time for this in your area. You may want to consider planting another where it will receive some afternoon shade, which will offer some relief from overly hot temps and scorching sun. You should also water the tree more during these higher temps, at least twice daily if over 85. degrees F.

The browning does sometimes occur where it produces seeds, but it sounds like you are getting a lot of seeds on the plants, so it makes them look rather bad. They typically seed more when they are stressed. I would give them some fertilizer and keep them watered for the spring. The extra seeds may have been a result of the bad winter they experienced. Give them this year to get back into shape and give them extra TLC to help them do that.

Mature trees are more equipped to resist disease, pests and fuller, dense foliage than younger ones, so fertilizing shouldn't really be a concern for yours unless it is beginning to show signs of stress. As long as it looks healthy, I would simply let it be.

Arborvitae are not considered a species with invasive roots, and the plumbing should be safe. Make sure to consider mature size in selecting a variety. This article will tell you more about growing healthy and beautiful arborvitae: https://www.gardeningknowhow.com/ornamental/trees/arborvitae/growing-arborvitae-trees.htm

Is the tree showing any signs of a nutrient deficiency? Are there any fertilizers that are being added to other areas of the landscape (i.e. lawn) and, if so, how much? Many times, trees will tell you if they need more nutrients. A soil test will let you know for certain if anything is actually missing. In general, mature trees benefit from fertilization every two to three years. One way to fertilize a tree is to broadcast granular fertilizer evenly over the area under the tree, extending to the drip line. Be sure to apply the fertilizer to moist soil or water it in immediately after application to avoid fertilizer-burn damage to the roots. Use a specially formulated evergreen fertilizer when feeding your Thuja Green Giant, assuming this is the mature green gaint you are referring to.