There are a billion and one opinions on dog food - but for what it's worth, I've always fed my girls a good quality dried complete food. It has been made to contain all the nutrients in the right quantities for dogs; and is easy and convenient to use. Honestly, there is no need to add anything to it, just make sure there's clean fresh water available at all times.
She's pooing in the crate because she has too - she just can't hold it in any longer. A crate is not a guarantee that a dog won't poo - it's just that they'd really rather not do it there.
If she has been out all day with you, walking or not, I'd say she is wildly over stimulated and desperately over tired. A 12 week old pup, IMO, needs sleep in the same was as a baby does. I think she is too young to cope with being out and about all day with no chance for decent sleep; puppies need managing to ensure they eat right, wee and poo right and rest enough. They also need careful managing to make sure they don't overexercise their very wobbly bodies and joints.
If she's been out and about all day for a week and has only just started taking in the big outside world, I'm not too surprised her poo has gone to pot. She's had too much to cope with; do the change of food thing and calm her life down. She's only been alive for 3 months - be gentle with her!
I really would drop the egg and chicken and feed dried food, three times a day. Get her into a little routine for a while; make sure she sleeps enough during the day and see what happens.