Animation is a large and complex subject, so I can appreciate how daunting it may appear, but I would recommend that you research the fundamentals of how CG animation works.
Typically rigging and skinning a mesh involves placing bones/joints where movement is required and then manipulating the amount of influence of each bone on the surrounding mesh vertices (a process known as weighting or weight painting) - the higher the influence, the more the mesh at that point will be deformed.
What you would effectively want is a bone for each of your flaps with the mesh vertices of that flap weighted to that bone.
After your mesh is rigged, you can then create a scene for each of your animations and keyframe the desired motions before saving and exporting the animation.
I’m a Maya user myself, and so cannot give you any help when it comes to using Max, I’m afraid.
Now, as for your point regarding whether these animations could be created in engine - yes, in a way, they can, though it may actually be more complicated from a programming standpoint, as you can actually manipulate skeletons within UE4. This can be extremely useful depending on your requirements. Things like turning the head to face a direction, or preventing a joint clipping through a mesh using collision are just two examples.
I highly recommend that you watch some of these tutorials involving “complex character animation” as they will demonstrate the processes that you need. You can just glance over the less relevant stuff.
Also, maybe ask your question on a 3D modelling forum too.
Sorry if you feel my answer lacks detail - but it’s difficult to gauge your current level of knowledge, and I don;t fancy writing down a full step by step guide. 
If you have any more specific follow up questions, then I’d be happy to try and provide an answer, but my main suggestion to you is to research and play around in Max a bit. Make some simple meshes and try to manipulate them with bones. You should soon reach a skill level at which you’re capable of creating what you currently require!