I just want to give you a giant hug and let you cry your heart out.
First of all, you are NOT worth less than you were before you started this semester! You are the same valuable, bright human being that you were beforehand. What you were trying to do is very, very tough - engineering is hard, chemistry is hard, chemical engineering is incredibly hard, and Purdue is one of the toughest engineering schools in the nation.
Lots of people who wind up becoming highly successful professionals have this kind of bump in the road somewhere along their path. You will still come out okay. You will get through the depression. You will get through the rough academic start. You will get through all of this! Things will get better, I promise you.
You need to take easier classes, in subjects that you truly love, to build confidence in your academic abilities, to keep your scholarship which is critical for you, so that you don’t have to live with your parent, and to continue moving forward towards an undergrad degree, in anything. Don’t worry now about what you’re majoring in. Just register for classes in anything that you love (check the reviews first to make sure that the profs are good, and that people can do well in these classes). Maybe take a literature class. You interested in history? Sociology? Art History? Music History? Political Science? Think about the classes that you had in high school that you liked - maybe take a class in that subject. The goal is for you to have a semester where you enjoy learning, get good grades, keep your scholarship, stay at Purdue (even though you might wind up majoring in something other than engineering).
I had this happen to me my first year at an Ivy. I was taking all premed classes (chem with lab, calculus, a biology seminar, a language class) and they were HARD. I had never had to work hard at anything beforehand, and it just wasn’t clicking for me. I wound up completely switching my major to something I loved (languages), got my degree in that. And a couple of years after I had gotten my degree, I decided that I still really wanted medicine, went back, finished my premeds, and went on to med school.
One of the beautiful things about our education system in the US is that we get a lot of chances at education. Chem E not working out? Switch to whatever you want - and you don’t have to make that decision right now. Try some other, easier subjects, and take it from there. If, after you earn your BA in something else, you decide you want to switch fields again? You go back and get a graduate degree. There are always second chances in education.
The thing that you need the most right now is a safe place to mature, and college provides that for you. It sounds as if taking some easier gen ed type classes for next semester, and taking a break from the STEM classes, is the best way for you to earn better grades, so that you keep your scholarship, stay at school, and improve your mental health. There is absolutely NO shame in doing that, for all the right reasons!
You could have a talk with your father tomorrow about the circumstances under which you would like to take him up on his offer of a free place to live this summer. Explain to him that you want to participate in the fun end of year school activities, that you want to go places with your friends. Assure him that you’re not drinking, drugging, or engaging in risky behavior (code words for having sex), but that you are a young adult who deserves the freedom to live their life without restrictions. If he agrees, give it a try. If he doesn’t, you cannot go live with him this summer - it will be a disaster. Instead, look for a very cheap off campus sublet, and I agree with working as many hours as you can this summer, to bank as much as you can ahead of time. Also, spending some time seeing friends will help to raise your mood. I think, from what you’ve described, that your depression is situational, meaning you’re depressed because you’re killing yourself trying to succeed at something very challenging, and it’s not working out for you. If you weren’t under this tremendous stress, I think you’d be okay - after all, you had fun on vacation, and you’ve made friends.
I don’t think that you should take summer classes, unless you have funding for it, and if you do, I think that you should only take easy classes in subjects that you really love. Please don’t try to take more engineering/STEM classes this summer to try to raise your GPA - it may only backfire. You need a break from the stress of the very tough STEM classes, and you need the confidence builder of some good grades in gen ed classes.