Oh yes, it does get better. I can barefly remember how hard it is with such a tiny baby, but you are doing a wonderful job meeting her needs! Believe everyone here when they say it does get better. Yes, she is trying to figure out day and night, and she probably is comfort nursing to fall asleep, not to mention sucking to set your milk supply.
The most thing in the short term is to understand this is temporary, and also do what you need to do to take care of yourself -- by which I mean make sure your latch is good. If it is a good latch, you can let her nurse forever and you won't mind. Seriously. But in those early days, I remember how hard it is, and how sore your nipples can be when you are new to nursing, and how sleep deprived you feel! For example, you may want to consult a lactation consultant just to be sure. Everything was fine with us, but I had a little soreness and that shouldn't be happening ...if it is, check it out!
Night v. Day -- It will work itself out in due time, but in the meantime, be sure to expose her to natural light so that she gets her internal clock reset...cosleeping is key --- you two can then lay in bed till noon and it doesn't matter. Gradually she will reset.
I remember my daughter did almost exactly what you mention when she was tired. What seemed to help was for me to nurse her, and then for us to use some kind of walking/bouncing method to get her closer to sleep. We swaddled her and bounced while holding her on a yoga ball, walked with her, whatever we could to get her to stop crying. My husband was key in this part, he let her suck his finger and then bounced, walked, whatever. We would just keep this up until she fell asleep, or sometimes I took her back and she was ready to nurse to sleep. I also didn't appreciate the importance of making a sleep-friendly environment. Turn off the lights, maybe turn on soft music, don't talk (whisper to your husband if you need to). All this will give her the cues it is sleeptime.
Also -- I think there is something magic about sucking and motion-- for me it was nursing and rocking, or nursing and walking. That was the magic combination that did (still does) guarantee sleep for my daughter. It is probably something that is hard to get the hang of with a newborn, but just keep the concept in mind.
My daughter still nurses to sleep and she is 14 months old -- before that freaks you out, know that is the way I want it, and it is because it works well for us -- and that after such a long time nursing, it is great. Actually, it is wonderful to know that at the end of a long day, all I have to do is sit in my chair and nurse her and rock her and she drifts off to sleep. Ahhhh...Don't worry...you'll get there! And to put things in perspective, it takes an average of 30-45 minutes from beginning to end of the process to get my daughter from fully awake toddler to sound asleep. Sometimes it helps me to look at a clock and tell myself not even to worry, just keep nursing and rocking till whatever time. It always works.
Hang in there! In the meantime know you are doing an amazing job of caring for your little one!
Babygirlsmama