Cut/trim to ground level or leave stalks?
Your goats beard will send up new stalks this growing season so go ahead and cut the old ones down.
Do I divide the plant for propagation, or do I have to collect the seeds. If I use seeds, do I need to dry them, etc. first?
You can divide goatsbeard in early spring. Look for an "eye" and try to include at least one in each division. It is also possible to grow from seed using freshly collected seed.
no other question
Sure! This will be best, as you are at the very high end of where they will survive. Protection from the summer sun and heat will be very necessary for their survival.
Here is an article that will help:
https://www.gardeningknowhow.com/ornamental/flowers/goats-beard/goats-beard-plant-info.htm
I transplanted it in June and it was doing wonderful. All of a sudden three days ago I noticed that half of the bush was turning black and dried up and on inspection all the leaves are dry and brown and the bush is dying. The only thing I can think of is that the gardening people who sprayed the lawn for weeds may have gotten some on the bush....or is this something else that has caused my bush to suddenly die.
Environmental problems for goatsbeard include drought or drowning in oversaturated soils; both can result in death of the plant. If neither of those are probable, it could be herbicide drift. The right site is important. They need well-draining, moist soil in part sun.
Also, its fall foliage is gold and brown.
Here is more:
https://www.gardeningknowhow.com/ornamental/flowers/goats-beard/goats-beard-plant-info.htm
https://www.gardeningknowhow.com/plant-problems/environmental/herbicide-plant-damage.htm
https://wimastergardener.org/article/goatsbeard-aruncus-dioicus/
My son is getting a new roof on his house. Unfortunately some old shingle fell on his goatsbeard and crushed it. If he cuts it back will it come back this year?
Yes, it should regrow this year.
https://www.gardeningknowhow.com/ornamental/flowers/goats-beard/goats-beard-plant-info.htm
These plants are in containers. This is the second years of planting. This year's foliage was much better and the blossoms GREAT. But, the plants died back in mid-July and did not grow back...leaving empty plant containers. Is there something we should do to continue the growth? Thank you for your time.
Goat's Beard prefers soil that is evenly moist, especially during the summer. Allowing the plant to dry out, especially in the hot summer months, will cause the foliage to decline. You can move the pot to a shadier spot with adequate irrigation over the summer months or possibly plant a friend alongside that is showy in the summer and fall or a plant grown for it's foliage such as coral bells, ferns or hostas.