Very interesting. I never thought of using cryptocurrency in this manner. The best wallet is to use your own credit card because no one else will help you...
You will always pay a transaction fee when using the bitcoin blockchain. This fee is used to pay the miners (nodes) on the network for continuing the blockchain. However, there are some wallets that charge a fee for using their wallet. I would look into using a hardware wallet such as Ledger, where you can keep your cryptocurrency offline.
If you don't own the private keys, you don't own the cryptocurrency. Websites such as Coinbase or Exodus so not allow you to see your private keys. Meaning, you do not really "own" that cryptocurrency.
Sort of like how a bank works, where you might have $3,000 in your account. But that $3,000 is not physically there.
