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I didn't get an answer to the bugs or if the Bugs Be Bone is dangerous to my health when it's sprayed'

Unless you are planning to cut them to the ground for restorative pruning, this would not be a good time for a trim.

Lilacs bloom on old wood, which means the buds formed last season, so to prune now would cut off the blooms.

Wait to prune till right after the blooms fade for the year.

This article will give you more information: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm

It doesn't look like snail damage, but it could be a large beetle or weevil or a caterpillar. Weevils and beetles can harbor in the soil or other habitat and come out at night to feed. Look closely at undersides of leaves and on stems for caterpillars, which can be camoflaged with coloring or hiding. Going out with a flashlight at night is sometimes necessary to find the critters. But insect pests would not break branches and twigs, so bird damage seems most likely.

Only thing you can try with pigeons is an owl or hawk decoy as a repellent, rubber snake, or a motion detector on an impulse sprinkler to spray when birds come in.

Bird netting over the shrub might be useful.

Neem oil spray is a botanical insect repellent, but needs to be reapplied every week during peak infestation period.

https://www.gardeningknowhow.com/plant-problems/pests/animals/controlling-birds-in-gardens.htm

If your lilac experienced a particularly harsh winter, it may be damaged leaves. However, it may have developed a disease. Inspect the leaves and stems for powdery mildew which is common in lilacs, and doesn't necessarily need treatment.

Here are a couple articles that may be helpful: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/common-lilac-problems.htm https://www.gardeningknowhow.com/ornamental/shrubs/lilac/lilac-care-growing-and-planting-lilac-bush-plants.htm

Cutting back, completely, is one common method of regenerative pruning. It can be dealt with using a less severe pruning method, usually.

This article will help you to know when and how to prune: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm

When cutting to the ground, the earlier you do it, the more time the shrub has to regrow during the season. Waiting till after they bloom can put you into late spring. You may not get blooms for a few years, however.
A less severe approach is to remove 1/3 of the oldest canes each year for three years. Then you should be able to enjoy blooms each year.

https://hortnews.extension.iastate.edu/1993/2-10-1993/lilac.html

https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm