Can a bow string be wore out and still look like it is in good shape?
Not sure what you mean, but normally no--it's either worn or not. The only exception I can think of is one that's been stored a long time and dry-rotted.
You can have a strand break underneath the serving and not notice it.
It can be worn out under the serving, with strands breaking but you cannot see them because they are under the serving. This can happen if you have exceptionally tight nock fit on you serving. If this was to happen, the rest of the string would look fine. If your string all of a sudden started stretching, I would consider this a possibility.
Bisch
Sort of like a guitar string. The string looks fine, but after a while it looses it's twang, and is hard to keep in tune.
As with the bow string, would it loose it's cast?
replace it.
Strings are cheap; learn to make your own. I replace mine regularly and only use really good 'old' ones as back up.
Building your own strings is great fun and useful. I'd go ahead and get a jig and try it put. Plenty of online tutorials.
Sometime where the arrow nock fits on the string the bow string under the serving can start to break however, you will start to see a lot of wear on the serving in that area. Once a bowstring is served that part of the bowstring nerver gets waxed again and can start to break under the serving.