To clarify, rarity refers to the fixed rarity of an item, being Common, Uncommon, Rare, Mythical, Legendary, or Immortal. Every item has a rarity and it does not change. Quality refers to special cases where an item is differentiated from a normal version by something, such as being Genuine, Strange, or Unusual. The quality of an item is separate from its rarity and has no effect on it.

I'd think that putting quality designations in the corner and keeping rarity for outline would be the better idea considering every item has a rarity but only a small percentage would have any quality difference from standard.

Strange Orange and Genuine Green (and Unusual Purple) are used for strange and genuine (and unusual) Dota 2 items, respectively. None of the item rarity colors overlap any of those colors. The only ones that're close are the colors for Immortal and Mythical, which are similar to strange and unusual respectively, but when placed next to each other it's relatively easy to tell them apart.

Perhaps it'd be a better idea to have strange/unusual/genuine items have a separate inner border next to the outer rarity border to prevent confusion.

The simplest method would probably be to just duplicate the TF2 system of having item icon borders be colored depending on rarity, and then also having item rarity listed on the mouse-over details at the bottom.

Assuming that's what you mean by implementation.

If the support isn't there, then what method is dota-trade.com using for their item rarity information? Am I to assume they just manually entered all the information?

TF2B would be the best and fastest Dota 2 backpack viewer out there if it weren't for one serious problem: Item rarities are not displayed anywhere at all. It's a pain to have to use alternate means to find out the rarity of items you're unsure of, and I wouldn't think it'd be a huge pain to add that functionality in.