How is the price of an ASA token determined?

I believe you won’t find the answer to that question on chain. The price of a commodity is determined by how much people are willing to pay for it. The “willing” part might not have happened yet, and therefore there might be no evidence of that. Price oracles usually use a combination of sources.

Is it possible to tell if an account already opt-in to an ASA?

Hi, Is it possible to tell if an account already opted in to an ASA? Assuming that no ASA has been transferred yet and I can’t rely on ASA amount since it’s not accurate.

Is it possible to store private data on the blockchain?

Yes, that would work. Blockchains are great for synchronization of resources (and/or assets) across organizations and individuals with conflicting interests. One caveat - I would store the private data on a centralized database before placing the transaction on the blockchain. The rationale is...

With the exception of data storage blockchains (such as IPFS), I would not encourage you to store your data on-chain. Instead, use the blockchain to do what it was meant to do - proofs & synchronization. Store the hash of the data along with a URL for the actual data on-chain.