My 3 year old is insanely jealous of her 13 year old sister and appears to see her as a real threat and whilst they do have lovely moments a lot of the day is spent with her kicking, biting, throwing stuff and even spitting!?! at her older sister , I tell her sister not to retaliate that I will discipline her but she now doesn't even appear to care about any punishments/being told off and eventually her sister does retaliate, for example will throw an object back at her after its been thrown at her several times or will kick her off her leg when she keeps grabbing and pinching her on the legs and I don't blame her to be honest, it's not fair on her at all and I need to get on top of it.
In general her behaviour has been difficult recently , only wanting to eat sweets and having violent outbursts , hitting and screaming at the top of her lungs if she doesn't get her way and also the same behaviour when out at shops/supermarket and she doesn't get a toy/sweet.
Bit of background - 3 year old was not a fan of dad for the first 9/10 months of her life and this I think upset him a lot so from around a year he bought her affection with sweets/toys , she only has to say daddy I want and she has it , no one can take 'her seat' on sofa next to him not me and not DD13. She also had a younger sibling born this summer, so this may have unsettled her/caused some of this behaviour , she generally seems to like the baby although there is jealously with her dad and he will limit the time interacting with/holding the baby whilst she is awake because he doesn't want to upset her.
How should I approach this because it's starting to feel like we are all being ruled by her and her outbursts, Any advice , thanks!