When you purchase an item through PayPal that is linked to your bank account when PayPal received a refund does the refund automatically transfer from your PayPal account into your linked bank account?
No it does not. A refund for a payment paid by Paypal will always go into your Paypal "Balance".
You can however then withdraw those funds back to your back account but it has to be done manually. To do this go to "Add/Withdraw Money" then "Withdraw Money" and put the amount you would like to withdraw from your Paypal balance and then select the linked bank account you want the money to go to.