Or even if you don't care about recent transactions, at minimum you'll need to either own >50% of the hash rate and just make your own chain (good luck) or be online every couple minutes to find the longest chain's block hash value.

You don't need fiber internet but just a DSL modem with most-of-the-time connectivity to do mining for contributing to the network's safety, but doing it offline altogether indeed does not really make sense. (If you're doing it for the moneys then you probably do want to get information early (stable, low latency internet) so that you don't waste hash cycles.)

The coin is transfered to your public key. This is saved in the public blockchain. Air gap means here, that you store your private keys on a device without internet.