All the private keys and associated addresses already exist. You don’t create them, you find them. You can do that multiple ways, by hand, with software/hardware, or combination
Discussion
If they already exist, how can you claim to access one, if not ONLY by going online and signing transactions?
You can receive without ever going online. But yes to send you need to sign a transaction and send to mempool
How can you "claim" an address as yours without going online? Your saying someone can just pick random addresses and send a bunch of BTC to them, and anyone who derives that address can go "get" that BTC? If that's the case, can't more than one person try to claim it?
Theoretically yes. You don’t claim private keys and addresses. Theoretically someone could randomly generate your same private key. But there’s some sufficiently large ( like more than the amount of atoms in the universe or something?) number of private key pairs to where both of those things are, in practice, impossible
Also, I am no expert, but this is my understanding.
That's what I also understand, but, gosh... That makes me feel a bit squidgy about the whole thing...