Problems animating character.

I haven't touched Max in over a decade, but I believe Character Studio and biped (or whatever is in it now) should work OK.

Are you sure it's working fine in Mecanim (i.e., test the model/animations in a basic scene without any UFE stuff)?

You're probably aware, but mecanim requires a standard FK hierarchy in order to map the bones correctly.  I've had to rejig a lot of my custom skeletons in order to use mecanim features.  Personally, it's worth it, as it adds a lot of useful functionality in Unity.  UFE, being one of the biggest with requirements of flipping animations for player 2 side.

Anyway, if you like, you can post the fbx/max for someone to look at.  Another thing worth trying, is looking at the free models on Unity Asset Store or Mixamo and see how they work.  They're a good baseline for how your rig should look.