A.,
There is nothing you can do about the fact that your sweet daughter is sleeping on her belly. Once kids learn to roll over it is safe for them to do so, and sleep. And at the age of 6 months the SIDS rate drops.
I don't know what to tell you about the puffy eye.
I do have a few recommendations about the wet diaper;
You might try lining it with a sanitary pad, for extra absorption. There are also diaper doublers.
Sometimes going up a size works.
You can also try differenty types of diapers. I've been diapering babies for 7 years and have worked with hundreds of newborns and know that it's often the fit. Believe me, there is NO ONE diaper that is the best but rather a diaper that fits YOUR baby best.
With one family, when all else failed the best success came from adding a diaper wrap (like the kind that you would use with a cloth diaper.) This did the trick for the family I was working with, with an almost 100% success rate. You can often find the wraps and childrens consignment stores, thrift stores etc.
Good luck.
J.