I am also having a batch of eggs hatch, Akira--it sounds like the first couple of my guys were out a couple days before yours, and the first few just started eating a couple of days ago. Sometimes they seem to hatch a bit before they're really ready to enter the world. The first two that hatched for me were in a set of eggs that were connected; perhaps it's the case that had something to do with it.
If you keep them in small containers such as cups, and provide a more or less constant supply of food (as Joan said feeding twice a day for example) they should start to eat. Sometimes you can see the orange in their system before you actually witness them eating, so it's a really good sign.
As for the cysts, I have a similar problem, perhaps moreso since I usually use ~60% hatch rate cysts (the nauplii/$ rate is the best). What I normally do is follow the method of draining from the bottom to remove most of the hatched shells. Then, I suspend the filtered result in water and shine a light above (in my case, I guess it's probably more accurate to say at that stage it's coming from all around)--in something like an old spaghetti sauce jar and wait for awhile. The unhatched cysts sink to the bottom along with any nauplii that have died. Perhaps a couple of unhatched shells float to the top. I use a turkey baster to grab the actively swimming ones out of the middle.
I usually do have an odd one or two that gets in with the larvae, but so far this really hasn't caused any problems for me. I have read that both hatched and unhatched eggs can cause death, but haven't experienced any yet.
If anyone else has a better way of separating them, I'd love to hear it, too. I tried a method once of having a capsule in the water where I put the shrimp/egg mixture with a tiny hole for the shrimp to swim out--it worked pretty well, but I still got a couple shells in it and it was a lot more work, so I'm back to my current method.