5 year old concerns

Hi, for the last 2 years my daughter has had speech and language intervention, resulting in assessements which have concluded she is 'average' for her age. her speech is very immature and she still frequently mis pronounces words and gets her 'he', 'she' pronouns mixed up all the time. In addition to this, her tantrums are frequent, most of the time if she hasn't got her own way. Tantrums can involve her forcing herself to froth at the mouth, calling us stupid, crying, lying on the floor. She has an incredible amount of energy which we try and get her to burn off by going to the park, she also goes swimming but she never seems tired and every night is a fight to get her to bed, despite bedtime routines, reward charts etc etc.

I hate the thought of her being unhappy and feeling like the odd one out but she displays no remorse over her behaviour, every day is like ground hog day in terms of her behaviour. She also displays no empathy whenver anyone gets hurt or is upset, she's more interested in the visual aspect 'is it bleeding?'. She is also very unaware of what day of the week it is, when her birthday is, when christmas is etc. I know most 5 year olds maybe have no concept of time, but i would have thought by now some elements would be there. She isn't at all tactile either, even when she wakes up in the morning, she won't say hi, or good morning. Very limited with her hugs too.

Sorry for the essay,  but i guess we really need help with managing her behaviour at home as it's affecting us all and making for a really sad house at the moment. We have had speech and language involved, and a child psychologist and we have our first ASD assessment next week. If she is diagnosed I am wondering what the next steps might be for us. Thanks for any advice.