If a mum has eaten a very healthy fruit, salad and veg based diet and eaten lots of gut-friendly foods like yoghurt and kefir AND also had a super-healthy pregnancy - then then the baby will probably not need a probiotic as they should have a pretty healthy microbiome.
Most kids who are rough or learn to hit have been in an environment where they have seen it and have learnt to copy it. If they see adults reacting calmly to a tough or demanding situation, they too learn to manage their emotions accordingly. Our reactions to children often set the tone for their learnt behaviour.
An expert article by Sarah Norris on how to be a confident parent.