This is an interesting idea!

What actions can the various players take, though?  The goals are clear, but I'm not quite sure how they're accomplished.

Also, about how long were you thinking each round would take?  A week or two?

Real-time battles, hopefully relying on a player's good judgement. Not sure whether to work in something like health - on one hand, it tracks your health, on the other hand can be a bit limiting at times and make the game too much like Bionifight, something I'm trying to avoid. Goal of the first round is just to survive, not sure how long. Second round ends when all Toa Stones are used (or destroyed). Third round, well, already stated what ends that. 

Thought about giving the Toa/Rahkshi/Makuta/Turaga some special moves they can use as well - about three each -, tied in with a non-regenerating 'Elemental Energy'. (All abilities use 5 Elemental Energy, all Toa and Rahkshi have 50, Makuta has 100, Turaga 25. This is replenished each round).

Like I said, it needs refinement. 

Possibly to make the first round less bland, the Makuta cannot directly act (also to prevent the Makuta somehow dying in the first round), but acts through six or so Kraata, with the goal of infecting Matoran for the second round. Infected Matoran then become Shadow Matoran, First infection is guaranteed and the chance of successful infection halves each time from there, and the Infected Matoran work to weaken the other Matoran and Toa for the second round?

I like the idea of a Matoran Nui, that would be a fun one. But there has to be some sort of risk / benefit thing to that. Like, before the round begins, all the characters have to say if they agree or not. If one of them is targeted for death, then they survive, but if two are targeted, then they all die. Or if one of the six becomes a Shadow Matoran in that round, then the whole lot of them becomes corrupt.