I would spend a day with an adviser at a community college near you. CCs are designed exactly for people like you, who have suboptimal high school experiences but want to get an education. There are high school graduates who have huge gaping holes in their education too. They also need to figure out the diploma vs GED thing, figure out how to place you correctly, give you advice on matriculation agreements, etc.
The rest of this is not to be discouraging. It is great to look at the moon and say, I will go there. It is foolish to build a paper airplane and figure out it is not the correct type of craft.
So keep your dreams - but maybe increase their scope - but also concentrate on how you will get there.
Also, you do not mention finances, but if you need to subsidize basic life, you may need to find FA that includes housing and other costs. This would allow you to concentrate on academics. Sooner would be better than later, avoid college debt … all the time.
Physics is really a very difficult major … but you could get there. Or you may find, as you take more classes, that your muse is really in computer science (gaming in an alternate universe) or science fiction writing or something different.
In Physics of the type you would encounter at Caltech or U Chicago, you need to be standing on the shoulders of giants, in other words, learning everything to understand Newton through Einstein through today. There are no shortcuts to get the knowledge you need to make a discovery or new equations or any thing like that.
Honestly both these programs would have such competitive students that unless you are super advanced in 3 years, well, lots of people stay away from Caltech … Someone mentioned a non-traditional program at MIT, and that is a much bigger school with more varied offerings.
Personally, I would work on a path of CC->state 4-yr (flagship if possible) -> top rated graduate school. Pitt is by no means not serious. I think the Comstock program at Smith or some other 4 year private school programs may also work well for you. A LAC might provide you a more customized education that can keep filling in blanks in your prior rocky education.
At CC, you will likely be taking some remedial, pre-college classes, but get an A in all of them and you have many successes - learning the material to high 90%+ level, proving you are an A student with lots of potential, moving closer to your goals, and … well allowing you to show doubters that you are the one who is going to go from no highschool to PhD physicist to household Physics genius.
And at some point, shed the “no high school education”
Physics and calculus are tightly related, so you need to get your math level up to Calc1 before you can take any college level physics classes. This one is easy, the CC will have a raft of tests to place you properly in the standard math sequence. SAT math is really somewhere around Algebra, Geometry level. If you balk at taking something called pre-algebra, just do fantastic in that, talk to the professor, and they can move you up at a much accelerated rate.
Finding new equations to solve math is probably not the way to learn anything. Again, stand on the shoulders of giants, people from Summeria on have developed math techniques that are efficient and fairly easy to learn. If you have a different way, great, but learn the normal way too. You need to have lots of tools in your toolbox and using a paperclip to remove a screw will not work nearly as many times as that boring screwdriver from Sears.
I would also start taking some non-calculus based, non-physics major physics. You need to work through some basic concepts.
Also have them place you appropriately in other fields. Your writing is college level …
And wikipedia is your friend. Bohr’s equations are from a different Bohr and refer to oxygen uptake in the lungs. The Bohr model of the atom was invented in the late 1800s. Bohr and Einstein do not contradict each other.
As the late 1800s date tells you, if you try to invent something at this stage, without being on the shoulders of giants, you will likely just reinvent the wheel.
The community college library or a local university library would have more enlightened passers-by. You should try to get a good reading list of physics books more on your level. Per wikipedia, the Hawkings book you were reading was deemed unintelligible by Hawkings himself (and no, this is likely not because of your creativeness, but just lack of knowing what you do not know). Popular science magaizine might be a good start, or some on-line science journal written for, well, below college graduate level.