Oh Cosmo, I really feel for you and I don't know if anything I write will help but its clearly early morning where you are and it's late evening here so at least I'm up.
Are you breastfeeding exclusively? Is there anyway that you can express and allow DP to feed baby? I know the cluster feeding is pretty brutual, my DS is 9 months and when he was born would cluster feed from 5pm - 11pm until he was a month. But by expressing and introducing a bottle it may help you get a chunk of a few more hours and you can work in shifts until baby settles.
Are you swaddling? I swore by the miracle blanket - we wrapped DS in this from day one only at night and he learned quite quickly that the swaddle meant night time which meant his long sleep (5 hours or so). I would also recommend perhaps practising the 5 s's, as shown in Harvey Karp's book, The Happiest Baby on the Block. You don't have to buy the book, just youtube his name to see how it works. Again, this depends on whether baby likes being swaddled but it does help with the wind because of the pressure on the belly.
You have NOT ruined your relationship, I promise you - pretty much everything you have written in your second paragraph is due to severe sleep deprivation, which is utterly destroying. The world feels like it is ending without much sleep, I know.
I remember thinking "shit, my life will never be the same again" when I had DS and you know what? Aside from the small loud little person that crawls around the house singing and trying to climb on the dogs, not much has changed.
Have you been out of the house much since you had your baby? It might be worth just heading out and getting some fresh air in both your lungs, I know the weather isn't great there but even a walk around the block will do wonders and will also help with your confidence.
I hope anything that I've written will help you and your DP - the first month turns everything upside down, not to mention the horrendous hormones. The night we got home with DS I sobbed uncontrollably for an hour after I got into bed. I felt so completely helpless in the face of this enormous task of bringing up this tiny, helpless infant. All I can say is that it does pass. Everything is just a phase. It will be so much more manageable when you get some sleep.
All the best x