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Yes, many people do trim back their fountain grass and other foliage type plants this time of year. It is not required, but it help to maintain a healthy look for the plants. This foliage will die back in the winter anyway. Yes, when pruned it looks just like someone gave the plants a flattop buzz cut.

Yes, you can go ahead and cut off the blooms but save any drastic pruning that it may need for late winter/early spring. This article will help: https://www.gardeningknowhow.com/ornamental/foliage/fountain-grass/fountain-grass-pruning.htm

Transplanting can be performed in early spring prior to new growth or after the growing season in late summer or fall.

I am not certain as I have not heard of this happening before, but I would recommend having the soil tested. I believe that the plants may be low on phosphorous or too high in nitrogen. This can cause a lack of flowering on other plants, which is essentially what happened here.

This artilcle should help with that: https://www.gardeningknowhow.com/ornamental/foliage/fountain-grass/fountain-grass-pruning.htm

Yes, it will grow back after burning and either method is acceptable. However, appropriate measures should be taken if choosing to burn and shouldn't be done when close to homes or other buildings. This article will help with pruning: https://www.gardeningknowhow.com/ornamental/foliage/fountain-grass/fountain-grass-pruning.htm

It likely was. There are several grasses sold as fountain grass. True fountain grass is not hardy in your area and would have died over the winter. When planting it in the future, make sure the variety of grass you buy is cold hardy in your area.