I recently started a hermit chunk and managed to get a grass block from an enderman and obviously don't want to waste it. Searching the internet for an answer has been fruitless.
Yes. Grass will spread from grass blocks to dirt in all biomes and all dimensions, including the nether
It will as long as a light source such as a torch beside the grass is there to allow it to spread. (as of Java 1.16.5)