Right, that appears to be one of the most confusing things in Tiled 1.0, so I'll definitely be changing the UI here, just need to consider the options. The thing is that when the collision editor is activated and you click a tile, then the Properties editor shows the properties of the Object Layer that covers the tile in the collision editor, unless you have an object selected there, in which case it shows the properties of that object.
So, I need to find a less confusing way to still allow the user to edit the properties of that object layer. On the other hand, I do not consider this object layer very relevant and maybe we could also remove that as a feature. Maybe it's something @Seanba could weight in on, because his Tiled2Unity is one of the tools best relying on the tile collision editor.
That means I need to fix the GMX export. Indeed it only creates object instances when a Type is specified, but in that particular place I forgot to take into account inheritance of the type from the tile. I'll make sure to fix that soon and include it in a Tiled 1.0.2 release. Thanks for mentioning it!