Try the spotlights/soint lights movable, they will give you the best shadowing system and will be dynamic. Use a skeletal mesh/interp actor for you mesh to be sure to benefit of the best lighting possible.
Using movable or skeletal mesh forces dynamic lighting, otherwise it'll use a lower quality preview dynamic lighting because it assumes you are going to bake the lighting later.
Skel/Interp actor use an Environment Light (visible in the properties of the actor), this a sort of cubemap that merge all the incoming lighting and compute the dynamic lighting on the object. Static meshes don't have these env light, also static meshes are good for props, not for character because lightmap are not designed to enhance correctly the details on characters.
Regarding you specular issues : be sure that your UVs are in the right orientation, Up to Down, not upward. Otherwise the UDK will compute wrong data. This shoulsn't affect the dynamic lighting however. Also, on a movable light be sure to disable the "Modulated" lighting, otherwise it will give you some strange shadows that you don't want for a turnable. http://udn.epicgames.com/Three/ModulatedShadows.html
And yes, if you can, add a physics asset to your character (no need to have a clean, you can use the default generated), this help to compute the shadow volume of your skel mesh.
Immovable lights cast per-object dynamic shadows (e.g. for skeletal meshes, interp actors). This is where a phys asset is going to improve visuals and efficiency.
Light Environments are an approximation of all lights (static+dynamic) affecting a skeletal mesh. This is important for in-game performance, for a turn table choose what's looking better.
I tend to use a dynamic mesh with light environment turned off so I get full control over every bit of lighting, and then enable high quality dynamic shadows on the moveable lights (light shadow mode normal).
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Right now my character is just a static mesh. Are you saying having a skeletal mesh instead would give better lighting/shadows?
About the physics attribute/volume, are you talking about having a collision object for the character?
Regarding you specular issues : be sure that your UVs are in the right orientation, Up to Down, not upward. Otherwise the UDK will compute wrong data. This shoulsn't affect the dynamic lighting however. Also, on a movable light be sure to disable the "Modulated" lighting, otherwise it will give you some strange shadows that you don't want for a turnable. http://udn.epicgames.com/Three/ModulatedShadows.html
And yes, if you can, add a physics asset to your character (no need to have a clean, you can use the default generated), this help to compute the shadow volume of your skel mesh.
Immovable lights cast per-object dynamic shadows (e.g. for skeletal meshes, interp actors). This is where a phys asset is going to improve visuals and efficiency.
Light Environments are an approximation of all lights (static+dynamic) affecting a skeletal mesh. This is important for in-game performance, for a turn table choose what's looking better.