It is late August and no blooms. The plants are heavy and healthy.
Back off the fertilizer if you are using it. There may be excess nitrogen. Also, they don't usually bloom till late in the season, so as soon as the weather cools a bit, they should burst into color. Here is general care info:
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm
I have nothing additional
Plant hyacinth bean once all threats of frost have passed. This is variable from year to year, but they can be started indoors to speed things up until ready to plant.
This article will give you more information on their care: https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm
Hyacinth Bean Vine. Can an existing vine be transplanted? If so, how and what soil? I don't have a picture.
They like to be planted where they will remain for their short life. They do not transplant well. I would recommend starting a new seed in the other area that you want it to grow.
Here is more information on the care of the plant:
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm
It's the end of the season and I'm anxiously awaiting the maturation of the pods to harvest for next season, and I noticed that most of the pods and a lot of the vine stems are covered with tiny, mite-sized bugs - should I be concerned and take any action?
It could be aphids or spider mites (but you probably can't see the spider mites). I would first try spraying them with water and see if they fall off. My experience with aphids is they just move out of the spray. I will get a paper towel and carefully wipe them off. Neem oil should help, too.
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/homemade-aphid-control.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/spider-mite-treatment.htm
Great advice - I sprayed the plant with cold water on the strongest setting and that removed them! Glad I didn't need to use any pesticides or chemicals.
Should I cut vine down from Trellis?
You can cut it down and save the seed pods to grow next year. The pods should be ripe about the time of the first frost.
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm
It is the best plant to grow?
Generally, you would plant during the day, but I suppose that you could plant one at night.
This article will help you:
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/hyacinth-beans-in-pots.htm
Do I cut it down after it is dead.
Yes. It will need to be replanted next season. This article will help:
https://www.gardeningknowhow.com/ornamental/vines/hyacinth-bean/purple-hyacinth-bean-care.htm