almost all Chinese nanny are paid in cash. So #1, #2 and #4 are not issues. For #3, you have to find another reliable one but use the current one as a backup if she is good.
1. Ask your nanny if she can provide her tax id for your tax filing. You need to cover her medical insurance if you want to file the tax. 2. Yes, if you file tax. 3. She may not be able to take a good care of your baby if she gets sick easily. Better pursue another nanny. 4. Yes.