There is a field to search for specific unique or set item (checklist).
Correct - currently only standard affixes can be searched, unique effects cannot be filtered.
No, you do not get the Favor back when you unlist an item. As you get reputation when you spend favor, which is when you list the item for sale, this would lead to a loop of list-unlist to farm reputation.
If there’s an MG Rank requirement on either the Exalted, or Unique item used to create a Legendary, that rank requirement is carried over to the Legendary. As it has a rank requirement, it cannot be sold.
It’s something we can adjust with feedback - though the power of Trade is not to be underestimated.