I don’t know enough about Mixer yet, so not sure about the UV, but my guess would be yes – UV unwrap your model first.
For your second question - since you need either an FBX or OBJ to import into Mixer, you’ll have to apply the modifiers first, then export the model to one of those formats, but you can still keep your original model with all modifiers. Depending on the modifiers you have on your model, when you come back to Blender you could still probably keep them since you are only creating textures/materials in Mixer. Any modifiers that wouldn’t affect your materials or UV maps would probably still be ok to keep. If you are exporting your model from Blender to a game engine after all the textures and such are done, then you would need to apply the modifiers that the game engine doesn’t support. Make sense?