Image texures are not critical you can simply copy the files to the exported .fbx and assign them in your target software. In Unity e.g. you would simply drag and drop them to your mesh.
Blender does export animations, make sure that you haved ticked all options in the fbx-export panel:
Put all actions on a single NLA-track, this shows the NLA editor with all actions I wanted to export:
Note that I left a gap between the actions otherwise the last frame overlaps with the first frame of the following actions. Also I avoided inverse kinematics since baking currently seems to have issues. (Can't bake single actions).
Importing in Unity proves that it works: