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The heavy annual pruning is done is early spring. Take about 1/3 of the oldest branches to the ground. This spurs new growth which is more colorful than older growth. Since it is a vigorous grower, you may need to trim the bush in June and again about now to keep it from getting too tall. Notice the difference between trimming and pruning. Vigorous growing shrubs need both.

These links will help you.

https://www.gardeningknowhow.com/edible/fruits/plum/growing-damson-plum-trees.htm
https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-pruning.htm

This really depends on the type plant it is. Some plants, such as bananas, don't like their leaves being plucked. Others benefit from it. Alternatively, the plant could be suffering from a pest, whether it be an insect, or a fungal/bacterial problem.

In most cases, it's best to avoid pruning frost-damaged plants during the winter, because the cut areas could allow more frost damage to happen. Wait until after the last chance of frost in the spring, then evaluate which parts of each plant are dead and prune off the dead areas.

This article will help you with pruning in the spring:
http://osceola.ifas.ufl.edu/mg/mg_faq_12.shtml#Freeze.18

Here is more about protecting plants from future freeze damage:
http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/protecting-from-winter-damage/

This depends on the tree, and your zone. Some trees are meant to be pruned very aggressively, while others cannot be done that way. Can you provide pictures and more information about the tree, and the area?

Spring bloomers like lilac should be pruned after they bloom and no later than July 4th.