Congrats on your new baby! Yes the exhaustion is unbelievable and no, you aren't unusual. (If you are currently wondering why anyone ever has a second baby or how on earth someone copes with twins when just one is running rings round you, yep that's normal too.)
Don't forget the whole day/night confusion thing, newborn babies are more wakeful in the night which might (only might) explain the crying (wanting to be awake and have attention). She may be sleeping nicely in the day because for her, that is the night time. Try making daytime more interesting - as pp said, take her out in daylight as much as you can, maybe do some baby massage, sing and talk to her, that kind of thing. Keep night time very calm and quiet and, well, boring. Yes, she will object and cry, but she will learn: nighttime is sleepy time. The day/night thing corrects itself in a few weeks, in my experience.
Every baby is different and it is definitely a trial and error thing, so keep looking on forums for different things to try. My DD quickly grew to hate her Moses basket, and loved being swaddled in her big cot. It felt cruel swaddling her but she was so cosy. My son hated being swaddled, and liked sleeping in a sling in the daytime.
Reflux is awful, my son had it. If the sick is regurgitation straight after feeding, like overflow, then keep your baby upright after feeding and maybe reduce the amount in the bottle a bit. Newborns have tiny tummies so perhaps a bit less milk more often would help.
Don't forget to wind properly even at night- winding a baby isnt a question of a little rub on the back for 30 seconds it can take quite a while! It took me a while to figure out my babies would ALWAYS burp if I winded them properly, and then they slept better. There are different techniques for winding, YouTube has some good videos. There are also some good videos about different kinds of cry and what is your baby communicating. I remember with DD I found it so hard to guess what she was crying about... but after a while the cries are readable. When my MIL annoyingly said, "oh there's a tired cry!" or "oh does your itty bitty tummy hurty wurty? Let's give it a little rub" it is possible she was totally right every time as cries do seem to have a sort of signature for different problems and you will learn to distinguish them (tired, hungry, bored, pain etc). Remember crying isn't always deep distress for your baby (although it may be for you) - it is communication, a language. It can be very satisfying to learn how to decode those cries in the months ahead.
Overall sounds like you are doing nothing wrong at all. Feeling inadequate and being sad is all very normal, especially soon after you've given birth. Hormones and pain etc. If those feelings don't improve speak to your HV.
And as pps said, genuinely ignore housework and take all offers of help you can get. Make sure you are getting some fresh air every day, and even if you can't sleep when the baby does, lie down and turn your phone on silent and rest.
Good luck, xx