Hi there,
Congratulations on the birth of your daughter, well done you! I am sorry things have been less than straight forward for you at this time. I'm not sure if you use this forum often, but you will find a wealth of advice for just about anything here, both through searching through old threads and starting new ones, like you have just done.
I too breast fed my daughter (still do), and like you, when she was little I was feeding every few hours through the night, sometimes more, not because I had to, but because she'd wake up and want a feed. Many, many small babies do this. It can be gruelling and exhausting, but it is a wonderful gift to be able to respond to your baby's needs in this way.
You don't explicitly say that you are exhausted, but you might be wishing for a little more sleep? As your daughter is happy to drink from a bottle, could you consider expressing milk and handing her to your partner or sister or mum, or someone else, so you can get a longer stretch of sleep, and they can give her the next feed of expressed breast milk from her bottle? You can get into expressing and freezing breast milk too, and then your partner could be almost self-sufficient in defrosting and making breast milk bottles for her so you can catch up on sleep. I found that if I knew that I would definitely get a long nap later on, I could always make it through to that time, however tired I felt.
It sounds as if you need to have a conversation with a health professional about your daughter vomiting up the formula you have been advised to give her. You could always speak with you GP as well as or instead of your HV, sometimes it is good to get a different opinion.
In order to strengthen your resolve to breast feed I would recommend seeking some extra support and advice about breastfeeding from other women who enjoy and like sharing their understanding of this aspect of motherhood, perhaps there is a local La Leche League group near you? Or a local breast feeding coffee morning where you can swap stories with other mums. It can be so comforting to know that you are not alone.
It is also common for young babies (and many older ones too...) to only want to settle to sleep in their parents' arms, and getting upset when put down. Your warmth and softness, the sound of your breathing and the beating of your heart must be so comforting to her, you can understand why your arms are her preferred sleeping place. My daughter was like this for a while; I would let her go to sleep while walking her around, and as soon as I tried to put her down she would get jarred and wake up. A friend suggested I turned laying her down into a bit of an art form, and it did work for us; I'd make sure the surface she was going down on wasn't chilly (we got a short shorn lamb skin and a super soft lambswool crocheted blanket (a 'christening blanket' from John Lewis, marvellous!), and then I would lay her down soooo slowly, keeping my hands underneath her for a while before very gradually sliding them out from under her.
Remembering that blanket has in turn reminded me that for a while I actually used it to swaddle her to sleep. It was amazing! When it was first suggested to me by my HV I thought it sounded totally archaic and mean, but my daughter loved it and was suddenly able to nap for hours at a time. I'd wrap / swaddle her, put her to the breast, and she'd be off to sleep, and I would be able to lay her down.
Some might cringe at the "falling asleep at the breast" scenario, but if it means she goes off soundly at a time when you need some rest, I'd urge you not to feel you can't let her do this. She is so little that it is completely appropriate for her, in my view, to enjoy this perk of babyhood. What could be lovelier and more comforting and reassuring than drifting off to sleep, snuggled in your mummy's arms, with a mouthful of warm milk made especially for you?
Do you have a sling? I would recommend getting one, so that you can still get around and do stuff, "hands-free", while she is strapped in, and getting the comfort she so enjoys from being close to your body.
It sounds to me as if you are doing an amazing job of making sure your daughter is happy and having her needs met, and I really hope you find the advice and support you want on here. Again, congratualtions, and take heart, all will be well xxx