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You can simply cut it back and then dig the plant up, getting as much of the root system as possible.

As long as the roots are alive, they will send up new growth. This article will help determine if the plant is still alive: https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm

This article will help with propagating the plant: https://www.gardeningknowhow.com/edible/herbs/rosemary/how-to-propagate-a-rosemary-plant.htm

There is a fungus on the leaves. It is probably pretty humid where you are which makes the problem worse. I would recommend when drying that you have a fan blowing gently over the area you are using for drying or try drying in a paper bag. This will help reduce the humidity so that fungus does not develop.

Yes, you can. This is called root pruning and is a common practice with root bound plants. When you repot the plant, make sure the container drains well and choose a pot about 1-2 sizes up from the one i is currently growing in. Here is an article or two that you may find helpful: https://www.gardeningknowhow.com/houseplants/hpgen/learn-more-about-repotting-houseplants.htm
https://www.gardeningknowhow.com/houseplants/hpgen/prune-roots.htm

It is likely just getting old. But, it could also be affected by a fungus. I would recommend treating it with a fungicide and see if that will help it. If it does not, then it may be time to replace the plant.

It is either underwatered or it has a fungus. If you feel it is getting enough water, treat the plant with a fungicide.