The short answer: imperfections in the clay.

The older clay had tiny amounts of iron residue in the mix.  Over long periods of time, the iron works its way to the surface and one sees a black speck.  It is said that those are actually brown because it is oxidized iron or rust.  They appear black because they are so tiny.  These are not considered damage but rather one prefers not to see them.

Should you buy a piece of bone china if they are present?  That depends.  If you enjoy the pattern, yes.  If the spots bother you from a beauty standpoint, no.

Does all china have it?

The better the brand, the less one is apt to see.  Shelley Fine Bone China, for example, paid more for their materials than most of the other brands.  Therefore they have less imperfections.  Also, the newer the china, the fewer the specks.

How should one clean old china?

If you are cleaning many pieces, place something in the bottom of the sink to protect them from bumping the bottom.

Use warm water.  If the water is too hot for your hands, it is too hot for the bone china.  Use lots of dish washing liquid.  Some people like to use a little vinegar as well.  That will remove tea if it has dried on. 

Then dry with a lint free towel.  If you allow them to air dry, they will usually show water spots.

In our shop, we do nothing else.  If there are serious stains, there are compounds and products for sale that will help treat them and sometimes they disappear.  In our experience, they often reappear again after use.  Since we don't want that to happen to our merchandise, we simply do not buy or sell stained china