Does the Rubik's Cube increase mental ability? Does becoming efficient in the cube help build your brain to be able to do other tasks better? What tasks and how?
Don't know of any research on the Rubik's Cube specifically. I did find this (https://arxiv.org/abs/1505.00750) but it looks like it was a presentation given by a mathematician at a mathematics conference, and it doesn't look like any actual experiment was performed to test the things that are being claimed. So I wouldn't take it seriously.

However, I would be surprised if practicing Rubick's Cubes actually did improve general cognitive function. Improving at specific tasks like brain-training apps usually doesn't improve general cognitive function (although evidence is still somewhat mixed), it makes you better at those tasks. Arguably, the mixed results can be explained by the fact that participants' performance will look look like an improvement in general cognitive function if you measure it in a task similar to the one that was practiced.

So in short, my expectation would be that it would generalize to very similar tasks, if anything at all. However, no one's actually done it, so if you try it and get results it'd be publishable, and if you want to form your own opinion the literature to look at would be cognitive training.