I think there’s only so much Monday morning quarterbacking that we can do here. I agree that these responsibilities sound like a lot for a 13-14y/o, on top of school… and school suffered as a result. But OP also found great value in being the one to step up for a beloved family member and doesn’t sound as if they’d do it differently if they had it to do over. The only harm it has done in the long run is an early blemish on the HS record. Perhaps schools that can’t overlook that are just not where OP needs to be. They are obviously a high-achieving student who will do just fine at whichever excellent college they attend.
OP might or might not have gotten into Rice, even with an academically-perfect freshman year. They might yet get in. They might not, and we’ll never know whether better freshman grades would have changed it. It’s water under the bridge, and OP has a bright future whether they attend Rice or not. They’ll almost certainly have multiple excellent options in the UC system (where freshman grades won’t affect the recalculated GPA). So, if Rice is the dream, OP should put their best foot forward in the ED cycle and hope for the best. But in the end, if it doesn’t work out, they’ll be no worse off than many, many equally talented and deserving students in the UC system, who aren’t at T20 privates because they either couldn’t afford it, couldn’t get in, didn’t see the point in paying twice as much, or actually liked the UC’s better, as many do.
Applying ED is smart because the application will get a closer read in a smaller pool, in addition to the “bump” in acceptance rate. Emory is likely a smarter choice for ED2 than Vanderbilt, unless Vandy is truly The Dream 2.0. Another option would be the top Canadian universities, which also don’t consider freshman grades (and which have fewer distribution requirements and more focus on the chosen major - possibly desirable if OP is very focused on their major path and research, and not as much on the breadth requirements of typical US/UC GenEd’s.)