Paypal - Website Payments Standard allows 'master', 'visa', 'amex', 'discover' cards on the paypal page do we have any option to remove amex from the listed cards via magento paypal Api? Is there any core hack. or If this was not a problem with Magento itself then I can go through the paypal settings to stop accepting the amex (But I haven't found any solution, still posting this as a question..)
If you are using Paypal Express or Paypal Standard, the card usage agreement is between Paypal and their customer. They allow their customer to pay for your product and then send payment to you. Under that venue, you basically have no say about what card is used since you do not gather or process credit card data. There are no setups in Magento to limit credit card type.
If there is a way to limit what cards you take with Paypal Express or Paypal Standard, it will be through your Paypal Merchant account settings. Visit the Paypal website, log into your Merchant account and review what settings are available to you there.
If you want to have a say as to what credit card is being used, you will have to upgrade to Paypal Pro and collect card data that you submit to Paypal as a payment gateway. You will then be able to select via multiselect which cards you will accept.