no, that's not normal. for right now? at 16 mos she can have milk rather than formula. add cream to her bottle. i have had to add olive oil to my kid's food, he's too lil. not too much or it'll taste iccky. weigh her often and keep track. if you're on WIC, they'll keep track too. she should, at this age, be giving you rabbit poops. with light smelling pee, if not, yup you're right, you need to check that out.
Definitely this. But also maybe slow her to feed herself even if its messy at home. I started doing that with my daughter and she started eating more. I only allow the mess at my home and she knows that but that helped me.
Becareful make sure she not lactose intolerant vomiting can be a sign of a food allergies