MB here. No, I don't think you're over-reacting.
The stove - she went against your instructions, so that's an issue, but I also always cooked on the rear burners when my kids were smaller and I think it's just common sense. I'm still pretty cautious, and I let the kids know why (they are 4 and 6). I let them stand on a stool and watch from a safe distance if they want, or if they are playing I make sure that they do so away from the cooking area. You need to make clear what your safety expectations are in this regard.
The bath - totally unacceptable. She should not leave a four year old unattended in the bath unless she has just stepped away to the next room for a moment or something like that. She certainly should not be going to the other end of the house and putting another child in the bath, and the fact that the older child was ill is even worse. I can think of so many ways this could have gone wrong. You need to tell her immediately that this is a serious safety concern for you. For some people this would be a firing offence, especially if there were other red flags.
Also, the excuse that she didn't want the younger child to get sick is ridiculous. Any experienced nanny should know that siblings and other family members catch things from each other and it is useless to try to prevent that. I just find her reason really strange.
Overall, if you want to give her a chance, I think you need to spell out for her what your safety standards are around things like cooking, bathing, driving, taking the kids to the park, crossing the street, when and where it is ok to leave the kids unattended in the house, etc. I wouldn't normally think it necessary to put this kind of thing in writing, it looks super controlling to say the least, but in this case I think it will help. You also need to send the message that if you ask her to take a certain route or only use the back burners, it's not optional.
However, I would be open to the possibility that this isn't the right nanny for you and start to plan accordingly.