We had a lot of this as well, it did pass..DD would vomit after most feeds.I think her tummy was getting used to everything.but in saying that reflux is very common and can sometimes need to be seen by a DR, if you think its more than normal. It could also be something in your diet if your baby is breastfed,thinks like oranges ,coffee, spicy foods can effect breastmilk and cause reflux.These are just the common triggers in food but it could be anything , you may want to do a elimination of some things to see if it has any change

With the sticking fingers down her mouth, its normal its all apart of learning , my DD did this she would gag as well,sometimes make herself bring up her milk, but it also passed pretty quickly once she learned it wasnt very nice

sorry typing one handed -

Oh and with the awful smelling 'old' milk vomit..i had many of those on me ,they really smell, but she doesnt do those anymore either