Would my finances be better long term if I just went to my state flagship and graduated on a full ride? I could do so within about 1-1.5 years.
A full ride is great.
Definitely apply for that.
Do run the NPCs on the colleges listed before (MIT, Cornell, Caltech, Stanford, Olin, UIUC, Cal Poly SLO, Georgia Tech, Purdue as well as the international programs) to see whether they’re affordable because they’d offer greater depth of learning with challenging peers and broader opportunities.
Affordable when there’s a difference in what you get and pushes you much further can be better than free that doesn’t necessarily get you where you want to go.
Graduating in 1-1.5 years is a recipe for disaster because for grad school you’ll need industry experience (internships or co-ops) and research, preferably presentations and at least one solid publication.
If you have a full ride, you should use your already-advanced classes to interview for internships all three summers, study abroad on a field-enhancing program, take graduate-level classes, get involved in research and defend an Honors thesis (this will be expected for grad school).
Obviously you should not go into debt for college.
So, run the NPCs to see which colleges are affordable.
Would I be a better engineer following the original pathway than the modified one? I ask this mainly for the early stages of my career, where it really matters just how good I am.
I don’t understand your question. If you mean your 2 HS pathways, you’d be a better engineer with path 2 (or path 1, but modified as per upthread).
How do I approach grad school if I do option one (and in general)? What are they looking for and which of the schools I match with would give me the best opportunities to continue my education?
Your grad school options will depend on your GPA, a GRE score, recommendations (so you’ll need to develop relationships with professors), internships/professional experience, research (conference presentations, publication, thesis). They will look for 3+ years where you’ve been challenged. This can be done by double majoring + honors college at your flagship though the above universities would have more opportunities for you as long as you do well there.
Is which graduate school I go to more important for becoming a better engineer than where I go for undergrad?
Your foundation is undergrad.
Grad school is specialization.
So, your college will give opportunities to become the best engineer you can be and the more aptitude you’ve shown the more opportunities.
Grad school will not make you a good or better engineer, it will make you more specialized.
Where you go for grad school does matter.
Is where I get my undergraduate degree significant in terms of content learned when comparing top schools to top flagships? In other words, is it that a student who goes to MIT is a better engineer because of MIT or better because they were among the top 0.00001% of high schoolers who got into MIT.
Because they have been deemed exceptional students and future engineers, they are given exceptional opportunities which in turn turbocharges their learning, experiences, etc. If you’re comparing UOk or Mizzou to MIT or Stanford, they’re very different in a out everything even though all are ABET accredited.
Do employers care more about where I got my graduate degree than where I got my undergraduate degree?
Yes, but they will also care what experiences (internships, job, research, volunteering, etc) you’ve had before grad school.