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This article will help with that: https://www.gardeningknowhow.com/edible/herbs/rosemary/pruning-rosemary.htm

Before you can save it, you need to know what's wrong with it. First , think about any changes that may have happened around the plant: have trees grown up around it to shade it? has something happened to change the soil around it so that it's staying too wet or too dry? was herbicide used in its neighborhood that might have blown or washed into it? Look at the leaves and stems to see if there are strange colorations or markings - these could be indicators or disease or pests. This article might give you some ideas: https://www.gardeningknowhow.com/edible/herbs/rosemary/brown-rosemary-plants.htm
If you see signs of disease or insects, you can take some leaf samples to the Extension Service for analysis. This link will help you locate the nearest one: https://www.gardeningknowhow.com/extension-search/

Cut! Cut, cut, cut. This article will tell you how: https://www.gardeningknowhow.com/edible/herbs/rosemary/pruning-rosemary.htm

Rosemary is really a zone 6 plant. You can get it to winter over in zone 5 if it is in a sheltered location, like near the foundation of a house. Even then it will be iffy. It is best to either bring it in for the winter or buy new ones each year. This article should help with its care indoors: https://www.gardeningknowhow.com/edible/herbs/rosemary/grow-rosemary-indoors.htm

H-m-m-m, I love rosemary. I can smell it just writing the word. It's always advisable to place a new plant of the same variety as one that died in a slightly different spot, in case there are lingering pathogens, and also to avoid possible mineral depletions. I should think a couple of feet away would be enough. FYI, here's an article on growing rosemary: https://www.gardeningknowhow.com/edible/herbs/rosemary/growing-rosemary-plants-rosemary-plant-care.htm

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