Hey all, sorry I haven't been active here so far. I've got a couple ideas, I'll throw them out and see what you guys think.
I love the Teddy Roosevelt idea, maybe we could even name the school after him? Theodore Roosevelt University (TRU) has a nice ring to it! For exact location, while Oyster Bay doesn't work, both North Hampstead and Huntington are close enough where I'd feel comfortable claiming TR.
Nicknames are where it gets a bit trickier, but my suggestion is Trustbusters. TR, of course, was known as the "Trust-Buster" and we could lean into some unique imagery. (In fact, I have both a TR and teddy bear logo churning in my head.) It bends the rules of "Status" a bit, but I think it's ok.
Colors are tough. The gray is a bit too bold for my tastes, I'd prefer if it was either lightened or darkened to serve as an accent. So, I'll suggest this: we use it as our primary color. Gray as our primary color with orange and cream accents, like this:

Black serves as a tertiary color. Finally, the founding date. TR passed away in 1919, so I'm thinking early 1920s, perhaps 1923 to honor what would've been his 65th birthday?
I really like the ideas we have going, let's keep it up!