Hey Time hope tonight is a bit better, it's so hard - have you found the 'high needs baby' support thread on mumsnet? I think it's still kicking around but can't remember what section it's in.... My ds1 sounds very similar, he was never content, had reflux, colic/wind and SCREAMED sometimes for hours, I used to think the neighbours would call the cops because he cried so much.
The thing that really helps is time (sorry) I'd say he started improving at about 4 months and if it's any consolation he was a very good toddler - mischievous but no tantrums really. I think part of it was he didn't like being a baby much - once he could sit, roll and grab things he was more content.
Gaviscon was useless for us, we used carobel thickener as he threw up a LOT and that helped. DS2 had bad silent reflux (he was prem and it's more common in prem babies I think) we were prescribed omeprazole by the Paeds at the hospital and that was really good. Is Calais getting any follow up from the hospital? If you think she's refluxy and in pain I reckon push for them (or GP) to prescribe something else for her.
Ds1 for a long time would only sleep being held. DH and I used to do shifts where I'd go to bed at 8 and he'd take baby til 12 then he'd wake me to take over until 4, then my mum (who was staying with us) would take him from 4-when he woke up for morning feed. It was grim but it did get better, eventually he would co sleep and then sidecar cot. He's 4 now and a good sleeper.
Long screaming bouts sound like over tiredness - hard thing is to break the cycle as when they are overtired they get all wired and hyped up and then can't sleep. Aidan would've slept in the car so I used to drive around for hours, or walk with the pram for hours just so he'd get at least two good naps a day. I also used to let him nap on his side or tummy (but never at night and I was always in the same room so could check on him) as it can be more comfy for refluxy babies.
I watched a lot of box sets and walked/drove a lot. Do whatever it takes to get Calais to sleep for a good long time and she will hopefully get less overstimulated and settle easier.
Sorry for the essay - my heart just goes out to you as I know how hard it is and anyone that hasn't had a high needs baby does not really understand what it's like. Still haven't read the whole thread, so just ignore me if you've already tried all this, or if I'm teaching you to suck eggs.
It gets better I promise! 