[color:"orange"] I believe you can do that. You need to click on the portrait of the character with Repair, then you should be able to repair in either inventory. I haven't used Repair, but it works for Alchemy and Sharpen. [/color]
Eum no. You can't. I've tried.
I know it works with sharpen and if it works with sharpen I can't see why it wouldn't work with repair, but apparently, life's weird.
[color:"orange"] Could be useful, although I don't think I'd like it. I don't want to repair all the random loot I'm selling. But each to their own. [/color]
Maybe not all the random loot, but you could do a repair-all feature where you repair all the equipment you are currently wearing.
Besides (I'm not a 100% sure about this, so don't shoot me if I'm wrong) I think all items you find come with max durability. So even if you would "repair" them it wouldn't cost you any gold.