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No, this will not hep with root growth. They need the tops to help build energy for the roots. If you are seeing too small of roots, the soil may be lacking phosphorous. You can add this in the form of bone meal.

Is the white powdery in nature or more like bleached in appearance? This would help us to better determine the problem. It could be attributed to either powdery mildew or from water spots associated with overhead watering or splashing of the leaves. Limiting overhead watering will help with this water spots. The powdery mildew can be treated with neem oil, which is an effective and safe fungicide. It will also guard against many insect pests. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

I would try adding some bone meal to the soil to give them the phosphorous they need to bloom and fruit. This article will help with growing beets: https://www.gardeningknowhow.com/edible/vegetables/beets/growing-beets.htm

This indicates a watering issue. Either they are underwatered or over watered. Unless you see standing water, it is unlikely that it is overwatering. Try increasing water to the plants.

It won't hurt anything. It is not not needed, but there is no harm in it.

It sounds like what you have is a root maggot problem. There are many species of root maggots so their color may vary from one to another. However, their taste for root crops, like beets and radishes, is the same. Here is an article that you may find helpful: https://www.gardeningknowhow.com/plant-problems/pests/root-eating-insects-identifying-vegetable-root-maggots-and-root-maggot-control.htm