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Your plan seems reasonable in that your list includes schools that you are likely to get into, but I think it's unlikely that Ivies will look seriously at your application with your GPA -- I am no expert, but think even the 3.8 would be a stretch unless you have done something unusual or your school has grade deflation, and the junior year grades will hurt your application. The high SAT scores will help you with excellent schools that are below the most elite level, but the top schools will view them as an indication that you were capable of getting better grades than you did. Hardships affecting school performance are best mentioned by a guidance counselor rather than being part of your essays, but I personally would be very leery of disclosing depression to a prospective school or employer. Parents permitting, you should look at private schools that you like that are low targets or safeties -- because of your high test scores, you might be offered merit aid there. Consider some of the women's colleges that offer excellent education and may offer a better chance at merit aid. You should use Naviance if you have it, or look at scool common data sets, and also look at results threads to figure out whether your college list is reasonable.
If you land at a state school, you can find a way to thrive there, and believe it or not, your parents' attitude about loans is very wise. Smart and capable students who go to state schools (and lots do) do as well with most careers as people who went to Harvard, and in some cases they do better. It's easier to get into med school if you are near the top of your class in a state uni than in the bottom half of your Harvard class. And many of your peers will spend decades struggling to pay off student loans-- or their parents will be unable to retire. Good luck with your applications!
Hated Gone Girl and Jody Picoult but I think my most despised novel of all time was "Lord of the Flies." Incredibly bleak view of humanity that seems to have been written mainly as a classroom (torture) exercise for high school students.
Here's the problem I have with this scenario. They sold the same seat to 2 people by accident. OK, so someone is going to have to go without. Why on earth not take the seat away from the person who bought it at the last minute, instead of the person who bougt it first?
Personally I would consider anyone who voluntarily did a 5th year of high school strange, no offense. You haven't been there long enough to realize this but you will outgrow high school long before you are done. AP courses are not equivalent to college courses; they are just massive, tedious, superficial surveys that represent hoops you have to jump through for competitive college applications. If scheduling conflicts prevent you from taking some, your guidance counselor should note that in his or her letter. And most rigorous courseload is a category, not a contest; it gets your application looked at, but it doesn't get you in. Every year there are students who are shocked that taking 2 years of college level classes with A grades (I may exaggerate slightly, but you know what I mean) was not enough to get into an Ivy. I'm all for taking an extra year to do something interesting and different, like study abroad, but an extra year of high school at home sounds unpleasant, unnecessary, and likely to make admissions officers look at you funny. And also, if you take a gap year, do it for you, not your resume-- apply to college during senior year instead of ruining a gap year stressing about how to get your applications done.
You should focus on finding and developing your interests, not on building an attractive college resume. Paradoxically, that is much more likely to make you attractive to an Ivy than 4 years of carefully manicured classes and activities -- provided you meet academic standards for admission. In any event, you should never set your heart on a reach school. Admissions to Ivies are a crapshoot-- you can be superbly qualified and not get in just because there are so many more superbly qualified students than there are spots in a class. And there are many, many schools that can give you an outstanding education and experience -- it's ridiculous to imagine that one school will somehow be better than all the rest, or even that you can make a rational evaluation of your dream school based on a few visits. I wish you luck, but I think you need to take a hard look at what you are doing.