I am building website which requires customer to update paypal account. Is there anyway to check the reality of customer's account? When my customer fill out their paypal account in my site, I want them to be directed to paypal login page to login and paypal will return the result. Does paypal api support this situation?
Pretty much any implementation of PayPal you choose would follow the flow you mentioned.
Payments Standard would allow you to create basic buttons or create an HTML form and POST directly to PayPal to process. It would send the user to PayPal for login and approval to complete the payment. The transaction details would include the payer status (verified or unverified) as well as the address status (confirmed or unconfirmed) and lots of other details about the order.
Express Checkout is basically the API version of Standard, but it's much more advanced and open to integrate in the way that works best for your site or application. In this case, some of buyer/transaction data is available during the process within your app through API requests and responses, and then you can also get to it via transaction details after the fact just like payments standard provides.
Another option would be to use Adaptive Payments, but if you're doing a general payment of any kind you probably don't need that. That's what you would use if/when you start wanting to split payments among multiple receivers within the same transaction, setup preapproval profiles, etc.
If you happen to be working with PHP my class library for PayPal will make the API calls very simple for you.