It sounds to me like your nanny is doing a great job, she just has different standards of cleaning. But if you really need your house to be cleaner then hire a maid or a maid service, it really should not be the nanny's job to clean house. If she is the onw making the messes then she should clean up after herself, but it sounds like she does, she just doesn't do a stellar job of it. One thing that you have to keep in mind is that she was probably not raised in the same germ-free culture that you were raised in and that she probably worries a lot less about germs and such. My parents had a Korean maid for a while and though she kept things looking tidy, they were nto alwasy sanitary, she would use the same washrag to clean the abthroom, kithchen, floor, etc. And she was mystified when my mom tried to explain why that was a problem! Anyway, there are a few siomple things that you can do to remedy some of your issues; here are my ideas:
1. Buy some disposable pull-ups for when the nanny will be caring for your son, you can save the cloth trainign pants for when you are home with him. This is kinder to the nanny and it will solve your problem with her not using the hamper for them.
2. Just learn to deal with doing the dishes yourself, she just has different standards of cleanliness and you are not likely to be able to change her mind. She is a nanny after all, not a maid. As for the counters, maybe you could buy her a container of Clorox wipes and tell her they are to use to wipe down the counters and they she can use as many as she needs and not re-use them at all.
3. Buy some hand sanitizer and put it in places where it is easy for the nanny to have access when she is caring for your son. Tell her to use it on his hands often and teach your son how fun it is to use it so he will ask for the hand sanitizer, buy a kind that is his favorite color or scented a smell that he likes.
4. You want her to organize all the toy? Seriously? That is asking a bit much. You can keep toys with lots of parts that you don't want scattered everywhere locked away and tell the nanny those toys are just for when you or your husband are home to play with your son. Make them special "Mommy (or Daddy) time" toys.
Remember that a little kindness goes a long way, she is doing you a great service by caring for your son and you are lucky to have someone who loves him and is kind to him looking after him. Even though she is not perfect she sounds like a keeper to me! Good luck.