For the chipped eyes: I have used a product made by Testors (they make paint and glue for model cars, airplanes, etc.). It’s called “Clear Parts Cement & Window Maker.” Clean the eyes, use a little painter’s tape to tape the fur out of the way, and smoothly cover the eye surface. Leave it to dry/set/cure completely without touching, which will leave a mark on the finished repair. I’ve used it on lots of Beanie eyes and they finish up glossy and smooth. I don’t know if the repaired eyes can be washed without wrecking the finish. By the way, I have washed many regular Beanies in the washing machine, first putting them in a lingerie bag or pillowcase to protect the eyes. Then air dry. If it is a large one, I may pick out a few stitches to remove the pellet bag first and replace it after washing. They come out great.
For the deteriorating fur, I have no solution except to just accept that it is part of the life cycle of a well-loved stuffed animal. You can think of it as the patina of love.