As far as a child coming to daycare sick I think all you can really do is give it a short amount of time and then send her home once you see issues. It is so hard to say if parents really are being sneaky or are really that disconnected from their child (or don't really care). If you call her enough times she will get the hint that it is just more convenient to stay home to begin with.

As for the napping - yes that is a big thing for me too! Try a different punishment for the child. For myself I will give a child a different snack in the afternoon for bad behaviour. So while the rest are enjoying cookies and milk that child gets vegies or a rice cake. I think that by calling mom you are telling her you cannot deal with her child's behaviour.