I had some spare time while doing some diskdump so I took a look on your mesh and it looks almost ok after tweaking it a bit:
+) Removed all those _end and _end_end bones. I would recommend to avoid to export leaf bones in blender.
+) Fixed the Restpose:
-+) Applied armature modifier to all mesh objects in Blender (so the modified pose would not change if the armature would be removed)
-+) Selected the armature object and switched to pose mode and did “Apply Pose as Rest Pose” in Blender.
-+) Finally choosed all mesh objects again in Blender and readded again a Armature modifier to all of them (and choosed the already existing armature).
+) Selected all objects (all meshes and the armature) and applied scale and rotation in object mode of Blender.
Exported from blender and imported into UE4. Retargeted free anim starter set and few third person anims. Result:
Export and import settings (edit: Export settings screenshot is from 2.75 blender but I used 2.76b which has more tabs. I usually use this settings for animations and leave all options enabled in the Animation tab of the export settings in Blender if I just export a skeleton mesh without animation):