Hi Solow,I am sorry you are going through a tough time.I can only reiterate what other posters have said.
Please please see your GP/HV and explain how you are feeling.It is quite possible that you have PND,and if you seek help things can become a lot more manageable with support
Additionally,remember you have only recently gone through the biggest lifechange ever and this added to the really awful experience that is sleep deprivation can really be totally exhausting physically and emotionally
Your baby sounds normal to me!!You are not doing anything wrong at all.Some babies are just more "High maintainence" than others.My first was a really placid easygoing baby,my second sounds like your dd in the early weeks,needing to be carried,and screaming if put down,not sleeping unless on me,so I hardly got any rest as I was scared I would fall asleep,roll over and squash her!!.It is so so tiring when you have a baby like this.I think on reflection I did have some PND symptoms with my second child but struggled on in denial,therefore missing all the lovely early stages of dds life,it felt more of a trial than a joy.
You sound like a lovely mum :
"I can't get any rest because I want to give her the best care in the world and monitor her well"
It DOES get easier as the baby gets older.I know at the moment that is probably not much comfort(If someone had said that to me with no 2 when I had had a max of 3 hours sleep for several weeks with dd I would have cheerfully murdered them!!).My dd is now 13 months and her and her brother are the biggest joys of my life.
I would suggest a sling so you can carry her and she is close to you.I also found a seat that rocked to be fairly useful.
Forget all the extraneous stuff,housework and the like.It doesn't matter for a few weeks,or if you can afford it hire a cleaner.I became a bit obsessive about stuff being tidy but wish I had ignored it all.
Finally make sure you look after YOURSELF and rest when you can.Keep posting on here it is a good support network.
Many hugs, Wrinkly.