Ok about the plants may I just say... I grew up in a house with a few fake plants, I really didn't know you had to water real plants! Stupid maybe but I never gave it a thought.
I think that even if the bedroom is on the ground floor, closing the door is just unproffesional. I view tv watching as a neccesary evil- 3yo is very tired and grumpy after a full day at school and needs 'passive' entertainment rather than drawing etc. But I use it only when I need to get stuff done or focus on the baby, who eats earlier than 3yo. I don't think this would be a final notice issue, unless you had children under 3, however it should definitely be raised.
When viewed with the final issue raised- thats maybe a final warning issue. Not so much the grabbing as sometimes safety is more important than how you ensure the safety, if I see 3yo heading into traffic I will grab her no mictake. However not telling you herself is terrible, either at the time with a quick text if appropriete or during handover. And as a live-in, there is even more time for her to have told you. I would have spoken to boss about it as soon as possible, not only to inform them of the two issues of a child heading into the road and my grabbing them, which probably upset them and hurt them, but also to cover myself in case marks did show or child brought it up. And presumably parent/carer of the friend over the road saw, what if they informed the parents?
Definitely needs a sit down, a good chat with no interuptions especially children. Have tissues to hand! Conversations like this will be emotional. Is there a three step dismissal and warnings clause in the contract? Its up to you if this constitutes a final warning issue, however you have to live with this person as well as entrust your children to her.... I do usually fall on the side of the nanny as thats my perspective but in this case I think you need to go with your instincts, if you can't trust her it may be time to reconsider the arrangement.