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It doesn't sound like thrush - that is an infection in the baby's mouth and your breast that is passed back and forth - yeast infection I believe. You would notice white patches in your baby's mouth, but it shouldn't cause so many other symptoms. You've only been nursing 3 weeks which could explain your sensitvities.
Newborn breastfed babies do have very loose bm's but they should be more yellow and seedy than other colors and mucusy. I think you need to check in with your ped about the baby's symptoms. I think it's always hard to know when you've crossed the line to diahreah with a breastfed baby since it's so similar, but the different colors and mucus are signs. The doc may do or have you do a smear of her bm's and have you keep a diary of your diet to see if she's showing an allergy or reaction to something you are eating. Usually you'd start with eliminating most popular allergens and then gradually reintroducing them while monitoring her bm's to see if any changes.
My little ones both went through "projectile periods" when they were tiny, but they didn't have the other symptoms of fussiness, gas, wheezing, etc. I have REALLY fast/strong letdown and so I think mine just ate too much in a very short time during the early weeks. They would "explode" at the end of feedings if I let them go more than 5 minutes, but they never seemed uncomfortable with the spitting. My ped said if they were content, then it wasn't anything to look into and to let her know if they got fussy or seemed to be uncomfortable --- which it sounds like your little one is.
As for the rash on her bottom...the thing that worked best for us was Aquaphor. It's similar to Vasoline, but the later didn't work for us, but the Aquaphor would clear up nasty rashes very quickly. My son's skin was extremely sensitive and we still use it on any skin issue he has and it always seems to help.