Hello,
You do not need approve the Request from the back-end as well, if you use PayPal, you have orders details in RealEstateManager component and you can see in Status what payment is pending and what completed.
So, you have details for every payment, you will be able find additional information about payment.
Test all you can with help Payment - Paypal plugin - you can find it in via Plugin in Joomla admin area
You need such filed in it: Real(yes) or a test(no) PayPal account: - this for testing.
If you not get Payment - Paypal plugin contact us through our contact form.
ordasoft.com/About-us/OrdaSoft.html
Thanks,
Taras
OrdaSoft team.