Lightning:
you'll need a lightning wallet for this task.
You send your funds, on-chain to the lightning wallet, then your lightning wallet uses this unspent output to create an on-chain transaction opening a lightning channel with an other node. From this point on, you can pay lightning invoices offchain. Once no more funds are left on your side of the channel, you can either leave it open or close it using an on-chain transatcion.

I've used zap wallet in the past, and on my mobile i use eclair


Wasabi & CoinJoin
Wasabi is a wallet that has built-in support for coinjoining. Coinjoining is usually fine privacy and security-wise. Other options are mixing or exchanging your BTC for a privacy-centered coin, and after moving those funds around for a bit back to BTC.
Each of these methods has pro's and con's, but they depend on your point of view... If you really want the highest level of anonimity, you'll probably need more than one method.

To be honest, i've never had my deposit flagged (so far), but i've heared horror stories aswell. Apparently, some companies actually do some data mining and follow the unspent-outputs-trace several transactions deep.

Using Bitcoin Core offline:
yes, you can create an address while being offline. An address is a hash of a public key belonging to the private key stored in your wallet. Your address is not "registered" with a central authority. If you create a private key, you can derive a public key, and you can hash this public key to generate an address. This address exists because you created a private key, not because it was broadcasted or registered.
It is possible to create 2 exactly thesame private keys, this is called key collision... The odds of creating a key collising when using a proper RNG are incredibly, incredibly small (allmost nonexistant).