Exporting from Blender into UE4 works well (static mesh, skeleton mesh with or without shape-keys/morph-targets). Exporting from UE4 into Blender does not work that well. It’s ok for a static mesh but everything with bones usually looks wrong. If you really need this direction (usually you don’t) then I would recommend Maya (LT) instead. Or probably just hide all those bones in Blender and replace the weird looking joint-shape with a custom shape. Don’t touch the rotations in Blender in Edit mode if you exported it from UE4. If you would create content in Blender for UE4 it’s easier to use a similar mesh (like from Makehuman or Manuel Bastioni Lab) and retargetting in UE4 (which works well) or the UE4 Blender addon from Lui - [ADDON BLENDER] UE Tools - Community Content, Tools and Tutorials - Unreal Engine Forums
That said - your first screenshot looks a bit …confusing to me. T-pose? If you would retarget to the UE4 mannequin animations your rest-pose in Blender should be (the UE4-mannequin) A-pose. The second disorted mesh looks like it got too less weight at some vertices (which usually could be fixed with normalizing the weights). Make sure there is no error and even no warning if you import your mesh from Blender into UE4. If you retarget something even take care that your bones are following the skelekton (except root, pelvis and the “IK”-bones that may stay animation). Like this: New project released by MakeHuman team leader - General Discussion - Unreal Engine Forums - Probably even take a look at the export settings: New project released by MakeHuman team leader - General Discussion - Unreal Engine Forums