R.,
First. Breathe in. Breathe out:) Being a first time parent brings alot of questions and anxieties about whether you are doing the right thing or not. I can't tell you what is th best thing to do for your daughter. Perhaps if her acid reflux is very severe she might need medication. Or I'm sure ther are natural alternatives that work just as well. Either way, just make sure you understand fully what is prescribed to her. If you think that her sleeping on her tummy would help, have you thought about letting he sleep on her tummy on your chest? Co-sleeping/bedsharing is a very personal choice and it has to work for you, your husband, and your baby so read up on the benefits and pitfalls of co-sleeping to see what would work for you. If co-sleeping isn't your thing, try bringing her crib in your room. You will be more in tune with her breaths and cries - even more so than with a monitor from the other room.
As far as her sleeping in her swing. There really isn't a time limit on how long is too long. If it seem to work for her...what seem to be your biggest concern about it? What other things are you doing that is working for her? If you closed your eyes and went within your wise mamma instinct, what would you tell yourself about what you need to do? If your baby could talk to you (and she is in her own newborn way:) and you stop and really listen to her, what would she tell you she needs?
At 4 weeks old, sleeping 5 hours at a time is quite a long stretch for a newborn...especially if she's nursing. I know you desire sleep...I completely understand:) But I'm wondering if she needs to be awakened and nursed more often..at least one time. Perhaps this will help her from being ravenous after 5 hours, which can cause her to eat too fast and possibly more than her little tummy can hold...which can cause fussiness after feeding.
Finally, you didn't mention if your daugter is spitting up alot after feedings. Acid reflux can almost be like a projectile vomit, and they cry out of pain. Is she doing these things as well? Maybe she just prefers to sleep upright. Maybe she's over full. Without knowing more details it's hard to say what could be some other things happening and to offer suggestions for things to try for her.
I'm sure you will receive some good suggestions from other mamasource moms. You can also contact a lactation consultant, La Leche League, and of course her pediatrician. Just make sure that whatever decision you make about this and every parenting issue, you go within and understand why you are saying yes or no to anything.
Best wishes on your parenting journey.
In peace,
A.
mom of 4. Birth and Parenting Mentor
www.birthingfromwithin.com