Thanks for the feedback! I can address a couple of things you mentioned.
First, I do have plans to add mouse controls. It may be a while though, as I will have to learn how to implement it. That goes along with adding touch controls for the future mobile version.
As for the arrow key behavior, when the cube is rotating, the rotation controls are disabled, but I don't think the face selection is, which would explain why that works while the rotation is happening. I don't necessarily see that as an issue, so much as something to be aware of, but maybe I'll look into it more.
On the ability to rotate the center section up/down/sideways, I have thought about adding that in, but I'll have to be careful I don't break what is finally working pretty well. Parenting/unparenting the pieces to the center piece of the rotation is tricky! You can rotate a physical cube that way, but it is also tricky.
Originally, the controls were designed with limitations in mind because of trying to fit the "Life in 2 dimensions" theme. I still want that to be an aspect of the game, but I also want to make it something people will enjoy playing, so I'm trying to find that balance. I have in-game tutorial stuff coming, hopefully by the end of this week, that will help those who don't already know how to solve the cube.
Finally, the leaderboards do actually work, i just wanted to tweak the numbers being submitted and displayed a bit.
Thank you for playing!