Not having a Written Statement is a bit of red herring I feel - a contract is formed when a person agrees to do work for someone. The contract at present is probably oral and may have some bits written in the form of email and text communication. I like many nannies would want the written statement before starting work as that is better than trying to remember discussions which may go back some weeks/months.
Without there being the written statement which would detail things such as amount of notice for taking a day off, then how would your nanny know how much advance notice you required. Yes I agree they should have Asked you, not Told you, so maybe communication between you needs to be improved, which may come about from you creating the written statement and having some way for you to communicate with each other whereby a record is kept (I suggest email and using a calendar).
The nanny should not call a child stupid but do you know the full circumstances of the event? People say things (often the wrong things) when under stress. Did your children say if they had done anything which may have resulted in the nanny saying they were stupid, or being stupid?
Given your children's history with nannies, are they in the phase of testing out this nanny, pushing their luck to see what they can and can't get away with? Could it be that having a word with nanny about being careful about the language they use may help?
I find there is always a time when the realities of the job hits home. Sometimes that is after a couple of weeks, other times it may be a bit later. Often though it is within the first two months - the novelty wears off and the job may not look as attractive it did initially. It takes time for everyone involved to adapt to the new situation. Whilst the children have had a nanny before, they have not had this nanny, so I expect they will be testing to see what is permitted and what is not, thus seeing at what point nanny gets cross. Nanny may be bit a confused about how pay is being structured, how much notice there needs to be on both sides for 'flexibility' requests.
So at the moment I feel this is coming down to communication issues and children pushing limits. You may or may not agree, after all you are there and I am not. Give it some more time, cross the t's dot the i's, do review chat (bad news sandwich style if necessary) to make sure nanny is happy with the arrangement and problems are brought to your attention quickly, resulting in quick resolution.