Can I firstly say, thank god I'm not on my own, and neither are you! You have just described the behaviour of my 5 year old darling daughter. First of all, the only things I would say, is a) don't go smacking - it doesn't work and they just throw it back at you during another tantrum (believe me, i've tried) and b) invest in an egg timer. It may sound stupid, but its one of the few things that actually (occasionally) help. I'll get back to that in a minute.
My daughter used to throw a complete wobbler once in a blue moon, recently, she's been doing it every couple of days! We are talking about neighbours staring at the house because they think she is being throttled, that type of tantrum. My husband and I used to have to open the living room curtains when she was kicking off upstairs, so that the neighbours could see we weren't up there hurting her.
We quickly worked out that the naughty step didn't work because she ripped the wallpaper off of the wall. Smacking just ends up with her thinking it was okay to hit you. I have tried threats to call a policeman, that doesn't work, I've driven past the police station and that only worked for three hours. Not sure if yours is the same but our girl can't seen to calm down, its like she's fixated on anger, and I am totally ashamed to say it but I was at the end of my tether recently and I had had over two hours of screeching, so I picked her up, put her in the bath and put the cold shower on her (for less than a second), to try and shock her out of it. It didn't really work because then she called me stupid for getting her wet, but when you are desperate, you'll try anything!
Like you, we chatted with her teachers. She hates change and is slow to settle but gets there eventually - didn't help that the teachers waited for seven weeks to tell us there was a problem though! But she's okay now and has caught up. She does get really anxious though and this leads to tummy pains etc, but I've found that I try and talk to her about things that are happening and not to focus on her tummy pains. I still have the days where you think, 'why can't you be normal and just do as you are told?' But I am rapidly learning that normal means different things to different people. I have gone down the route of thinking its ADHD and recently panicked about it being a mild case of Autism. My mum soon snapped me out of that one thankfully. Children are not stupid and infact are very manipulative, I've learnt that if I want my child to do something, I have to (rather annoyingly) make it fun or turn it in to a challenge, its time consuming, but then so are the tantrums.
Compromise is a word they quickly pick up, such as 'if you put your toys away first, then you can have a biscuit', and praise works a treat. I try and praise mine as much as possible even though you think 'do I really need to say well done for remembering to pull the flush without being asked' but you'd be surprised.
Back to the egg timer. It was suggested by one of the teachers to try and improve her concentration (she fidgets - name one 5yr old that doesn't). Anyway, we sat down and explained to her that she had to do 10 minutes of concentration time after school every day, such as reading, spellings etc. She picked this up well - home from school, biscuit, drink and then concentration time before TV etc. Well we are now trying to use quiet time e.g you can come back downstairs when you haven't screamed or shouted for 2 minutes. During a calm period, I sat with my little girl, held her hand and we kept quiet for two minutes and said that when mummy says you need to be calm for two minutes, its not really that long. Once the egg timer goes off, she knows that thats the end of it and we can move on. We are still in the early stages but she seems to be grasping that mummy isn't backing down and even though she has stamina that would put a marathon runner to shame, the screeching sessions seem to be reducing in length. Now all I have to do is try and get my husband to abandon his philosophy of 'anything for a quiet life' and back me up!
Sorry about the novel, and I don't know if it will help but its helped me by knowing that I'm not on my own, and given me a chance to offload.