My first slept with one wake up a night from 3 weeks... by 8 weeks she'd go down at 7 most nights (bear with me)... then we hit the 4 month sleep regression.
Our wake ups were like the ones you've just listed. It was brutal. I was laregly on my own as DH was rennovating our new house.
Now my problem was that I had always fed her to sleep, so everytime she woke up I had to feed (breastfeeding).
I realise that isn't your problem.
This carried on for 2 months (where I tried extra feeding in the day, adjusting nap times, dummies, blah blah blah). I waited until she was 6 months and then I did controlled crying. (Not a popular choice on here).
She is 4 now and she sleeps for 12 hours every night and if she isn't at nursery she will still have a 2 hour nap during the day.
I am not going to try and give you advice because I I don't have any (my second was a completely different story and didn't sleep well til he was 1 and even then it was hit and miss), but I do want to give you hope. Bad sleep now does not mean bad sleep forever.
In the meantime take ANY help you can get and don't be afraid to ask for it. Sleep whenever you can, even if only for a very quick nap (Google sleep debt). You will get through this.
My third is now 6 weeks so back to sleepless nights and I find I feel much better on days where I am out and about (though be wary of driving). A sofa day now and then is fab but definitely makes you feel even more tired and desperate to sleep.