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Chances are that it is slugs. They only come out at night and will typically strip a plant of its leaves. This article will help:
https://www.gardeningknowhow.com/plant-problems/pests/facts-about-slugs-and-how-to-kill-garden-slugs.htm

We sow rue seeds in mid January to sell in 4 1/2" pots at May 1. The seed are put in a germinating mix and placed on bottom heat, kept barely moist and in bright light (greenhouse). When the seed have germinated (1-4 weeks), we remove from the heat bed and let them get 2 sets of "real" leaves before transplanting into small cells/pots, about 2-2 12". We use a commercial soilless mix. And when the roots have filled that container, we bump them into a 4 1/2" for sale. DO NOT keep rue moist, rue does not tolerate overwatering. Let the plants get a little dry before watering. Once established in the ground, they may not need additional water except in drought. A January seedling usually grows to appx. 12-15" by early July.

This article will help you and also has a picture of rue: https://www.gardeningknowhow.com/edible/herbs/rue/growing-rue-herb.htm

Yes, you can cut off the dead or brown growth to allow the new growth to come through. This article will also help with growing rue plants: https://www.gardeningknowhow.com/edible/herbs/rue/growing-rue-herb.htm

The plants usually blooms in mid-summer. Here is more information about rue plants: https://www.gardeningknowhow.com/edible/herbs/rue/growing-rue-herb.htm

Rue should be ok. It is considered to be on the list that deer will eat only if there's nothing better. This article has info on growing rue: https://www.gardeningknowhow.com/edible/herbs/rue/growing-rue-herb.htm