My bank account is linked with my paypal account. I know how to transfer money form my paypal account to my bank account. What I am wondering is that if I receive money in my paypal account, does my bank knows this?
When you link a bank account to your PayPal account, all you are doing is saving your bank account information so that PayPal can pull (withdraw) or push (deposit) money from/to your bank account. In a sense, you can think of PayPal like a separate bank (even though they aren't truly a bank). When you deposit money into bank 1, bank 2 doesn't know about, even if you have set up the ability to transfer money from bank 1 into bank 2.
No, your bank does not.
No. Different financial institutions do not share data like that, for competitive reasons at the very least. Deposits into PayPal stay in the PayPal system until you do a transfer, unless you have set up something special.
If your logic is that the banks report transaction data to the IRS and PayPal differs from them by not doing so -- that is a wrong assumption. You should prepare for, and even count on, anyone involved in you receiving money reporting that income to the IRS. That does not automatically mean you pay tax on it; but IRS will be looking for some accounting of it when you file your 1040 at the end of the year.
The answer is yes, if you gave them permission to. To give them permission just give your bank the username and password, and accept the terms.
As stated in the question many financial institutions do allow you to easily transfer funds to and from other financial institution. But that doesn't mean they can see the balance of the account, or even any transactions.
But there are services that allow financial institutions to aggregate all your accounts. Your financial institution contracts with a third party who stores all your login information for your banking and investment accounts, and then when you go to the website you can see the account balances and transactions. I believe that PayPal accounts can be accessed this way. Many mutual fund companies do use this service so that you can see all your investment accounts at the same time.
This is a service that many people use. It is similar to what Quicken or Mint or any online service that downloads banking information does.
The responsibility of reporting income to the tax authorities is the responsibility of the "bank" that holds the funds, it is not the responsibility of the aggregator service or the "bank" that uses the aggregator service. But once you give the external login information to a financial institution, then your are potentially exposing your transactions to other financial institutions.