Hi there, @aricl. It looks like your character is sorting against those trees as I would expect. The problem is that those trees are made up of separate tiles. What you want is for your character's root position (at his feet I guess?) to always sort against the bottom of the tree. If you change your tree tiles so that they are twice as tall then it should work. Note that this is what the example that comes with SuperTiled2Unity does.
As for your second question that is more of a Unity issue. My advice is to get your collisions working as desired without using ST2U first (i.e. just using stock colliders). Once that is working to taste then you can take the lessons learned from that and apply that to your maps exported from Tiled.