(sophomore) As a result of personal issues, familial issues, health issues, etc, I’ve had an awful high school experience so far and pretty much ruined any chances of getting into a top school (MIT, Stanford, Caltech). I’ve since then remedied those problems (for the most part), however, I can’t help but worry non stop about how much my past mistakes are going to affect my academic future. I’ve done light research, and from what I can tell many of the top graduate schools have a vaguely nepotistic system of only admitting previous attendees of the same school into their graduate program. Assuming I excel in whatever undergraduate school I make it into (I have my sights set on Georgia Tech), how much will the school itself affect my chances?
If you do well at a school like GA Tech you will be fine.
Hopefully you’re not thinking of grad school in the social sciences 
Seriously, you are completely off base here. Do you think applicants to top programs are evenly distributed among all colleges? What you think your “research” shows is seriously confounded by student characteristics and choice. That’s good news and bad news. The bad news is the school doesn’t “get you in”, the good news is that if you work hard and get to know some profs for strong recs you can get into a top program no matter where you do your undergrad.
You are thinking a little too ahead of yourself, since you’re a HS sophomore. Focus more on the “now” aspects: doing well in HS, participating in clubs/activities you like, perhaps prepping for the SAT or ACT, and enjoying life.
Also, grad school admissions is not determined just by where you went for UG, but many other factors like GPA, research/work experience, professor recommendations, standardized tests, etc.
For undergrad -> PhD school, the strength of your undergrad in your major can matter somewhat, but a good flagship-level university like Georgia Tech will not be an impediment for most majors it offers. Obviously, as noted previously, courses, grades, research, recommendations, and possibly GRE matter.
Rankings in major of your PhD school may matter much more when it comes to PhD -> tenure track faculty hiring.