I got a very good book when my second was little which I think you'll find very helpful. It was so good I gave it to our local breastfeeding councillor for her 'library' of books. It's 'Colic Solved' by Bryan Vartabedien who is an American doctor. Available on Amazon.
I don't know if you had heartburn when you were pregnant. If you did you'll know a bit about how the baby is feeling. Keep the baby upright as much as possible.
Is baby breast or bottle fed? If breast fed there is a way to feed them so they're kind of upright. You lay them on your tummy, you're sitting up but leaning back a bit (so you'll need to prop yourself comfortably); you're tummy to tummy, babies head up at boob, feet down towards your legs.
Did the doc give you any medication for him like Infant Gaviscon? Mine had that but became constipated on it, but I've heard great things from others. There are other medications.
Try an osteopath that specialises in babies.
Are you sure he's constipated? What's it like?
Regarding getting things done during the day, forget it! You won't look back and wish your house had been tidier or cleaner - plenty of time for that later. If anyone else is bothered by it let them get on with it. You just have to get through this time and it's much easier if you just concentrate on the task in hand (baby and you) and keep to the present.
He doesn't seem to be losing weight which is good with such violent vomiting. Mind you, I was apparently a vomiter and was also a chubby baby.
I also liked the Happiest Baby on the Block by Harvey Karp (also an American doctor). The first doctor doesn't like this book, but I think you can use the second book for calming techniques and the first for help specific to reflux.
Re. milk intolerance, if you're breastfeeding come off all dairy (it can take a few weeks to see the difference), if bottle feeding there are dairy free formulas - ask doc for a prescription.
Be careful about the co-sleeping when you're so tired. Can you get a cot with a removable side? Put it on a slope with lots of books under the feet at one end. And try and swaddle him for the next couple of weeks (my baby would break out after 8 weeks).
Thrown a lot at you here - God I feel for you!
I hope the docs at the hospital gave you a bit more help than a diagnosis.