Our requirements for Paypal Integration seems customized from the Paypal's available products. Paypal documentation is a bit vague to figure out the differences down to earth.
I would like to choose between Paypal Payments Pro, Payments Advanced and Payflow Gateway or any other product that satisfies.
Our Requirements:
Payments Advanced seems to work for 1,2,4.
Payflow Gateway seems to work for 1,3,4,5.
Payments Pro is supported by Paypal Express Checkout, which is usually a redirection to Paypal Website. So requirement 1 is ruled out.
If a buyer pays with their PayPal Account they will be redirected to PayPal so they can log into their PayPal Account, this is how PayPal works.
If you want people to stay on your Website you need to get setup with PayPal Payments Advanced and they will stay on your website in iFrame.
PayPal Payments Advanced Getting Started Guide
Payflow Pro is straight credit card processing so no PayPal Account is used to process these payments.
Express Checkout is using PayPal so it does not work in an iFrame and it will redirect to the PayPal login Page by design.
Your other option is the embedded checkout design here is the documentation on it
PayPal Embedded Checkout Design Integration Guide