Software like this could help to easily animate the 2D elements you paint. Certain post processing effects are used as well as particle systems to create the real polish but otherwise it's the grunt work of 2D animation. The 3D look is achieved with the overlapping layers moving with a slight offset. Like the weapon in the foreground of the example you posted. Its broken into 3 maybe even 4 parts in front of the character's hand. Using dark areas to separate the areas and offsetting the movement by a pixel or two. The blink is huge, past login characters haven't had it and it looks terrible. The more layers the more complicated the workflow as you attempt to give the character a sense of weight and natural movement but the better it will look. Subtle movements will look better since your movement is limited to hinge and chain movement. But exploring a variety of transform, scaling, skew and perspective alterations can allow for a bit more flexibility. Photoshop enables a transformation called warp that could enable a lot of the bending you see in the video. I look forward to see what you come up with.
The game that kind of popularized this style of animation in my opinion is Homeworld and Homeworld II cut scenes.