If you give someone the password to your iCloud Account through a phishing message or even using the same password on a website, then it can certainly be used to compromise your account.
If your account has been compromised, then follow the steps here.
If you think your Apple Account has been compromised - Apple Support
If you are unable to regain access, then it will require you to submit documentation to remove the account using the method provided here to remove an Activation Lock. In most cases, that requires the original purchase receipt from Apple or Authorized Apple Seller.
https://al-support.apple.com/#/additional-support
A website changing the password of your account used on that website will have no effect on your iCloud account. The only way that would be a problem is if you did use the same Password and that should NEVER be done.