Actually a rubics cube is very simple to animate. And set up.
Just make a bunch of cubes in a array that forms a rubics cube. then place all their pivots to the center of yoru cube. Keyframe all cubes and set rotation interpolation fo teh curves to quats. (if you neglect this step your cube WILL break)
Then just key in the groups and animate forward (once youve set a step you can NOT GO BACK without having to redo all you do after this) by placing linear rotations in world space (so make sure your default tangent type is linear, and rotate tool in world space, Also you need to keyframe all cubes at each step so you wont go of sync, and never rotate more than 90 degrees. Tip: make a character set. Warning its VERY easy to do a mistake so THATs why its best to build simple scripting supports).
As for need of mel scripts well you can not get anything doen in maya without mel scripts, since the whole GUI is a mel script but yes you dont to wirte any scripts.
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1. Is it actually possible to animate a Rubiks cube without using a heavy Mel script rig?
Yes, assuming that you mean by melscripting not needing to write one. As i said its NOT possible to do ANYTHING in maya without mel scripts. So in way its not actually possible. BUt assuming your scope of thinking then yes.
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2. Is there a way to set it so that when I rotate an object it does not rotate the local axis along with it?
No that's how eulers work, but using quat interpolation solves the problem as the eulers set the end orientation but maya interpolates then the shortest way between theese points. Wich is what you want.
But yes it is possible to do the interpolation manually too.
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Well you should stop thinking that the scripting language is a must its a may thing you may do it. If you wouldnt then maya would suck big time, and frankly be near useless.