I'm not sure what you mean by factory sponsored assignments so here is a picture.
If you choose "Free Assignment" and then "Eden Fantasys sponsored" - you can one every 30 days as long as your prior assignments are done.
It is my understanding (and I'm trying to remember this for sure) that you can have one "at a time" of the manufacturer-sponsored
assignments as long as your other free assignments are done. So for instance, you have your free assignments for Eden plus any buyouts and they're still outstanding - then you can't request a manufacturer sponsored assignment. You must have all prior free ones done. That is what I think I remember but it has been a bit since I tried for one of those.
As far as buyouts go - shop on the site and on the product page use "grab it" near the top right of the page to put stuff in your assignment cart.
When you go to your review program page (under "account" on the top left side of the page) - you will see "assignment cart". When you ask for an assignment it will ask you if you're wanting a free assignment or a buyout assignment.
Put your free assignment that you've been given in your cart - then do any buyouts you want and add them to the order.
You should have free shipping for your order.