Cons: Could be lost or damaged physically, rendering the crypto useless (although some come with backup features now)
Always write down the seedphrase of your wallet and keep that in a secure location. If a wallet does not allow you access this, avoid it, cause you won't own whatever you store on it.
Wallets didn't store any keys, it's just a software to connect to blockchain.
If it's a closed source wallet, you never know for sure what information the wallet stores and which it doesn't.
Well, it's better to find open-source wallets. There are plenty of them. Cold wallets are not for everyone anyway.