Please help me!

My boyfriend’s mother recently gave me a kitten- she doesn’t know how old she is, but did say she was eating hard food, so I’m assuming around 5-6 weeks? She also told me she has been using the litter box, but little Ellie hasn’t gone to the bathroom at all since I got her, anywhere. My kitten has been eating, but just recently has began throwing up everything she takes in, including liquids. Any suggestions? I also need some help as to introducing her to my other pets- two dogs and a tomcat. She is okay with my Pug, after a day of hissing and howling whenever she saw him. I know introducing her to my Boxer will be another story all together- he is very curious and investigative, and doesn’t understand that he is bigger than everyone else and I’m concerned. I haven’t let my tomcat anywhere near her as of yet- he’s never been around another cat before besides one that lives next door, and he fights with that one endlessly. I am also concerned about her behavior- for example, right now she is lying in my lap looking up at me and purring away, and just a few minutes ago, she scratched and clawed at me to the point where my hands are dripping blood and I’m going to scar. I’ve had kittens that have scratched before, but not to this extent. She also starts howling out of nowhere and I don’t know what to do. My mom has agreed to give me a month to fix everything before I have to find a new home for little Ellie. Please help! What do I do?