It's the target's weapons, and the Dancing Rune's effects. Therefore, they're under the target's command because that's who'd command them if a Dancing Rune were activated.
Yes, their weapons start animating w/o the activate action because the Rod's effect first says the weapons are animated, then it describes in what way. The Dancing Rune is only a reference for an animating effect that the Rod started, not a potential animating effect that needs activation.
What's crazy to me is just how many weapons might activate!
Hmm... Interesting.
Do they start animating without using the dancing rune's activate action?
That is how I read it. Activate the rod and the rod activates the dancing effect itself rather than having the target creature need (or be able to) activate the rune effect.
Though I would not require that for the weapon that the target is actively wielding. That one the target can choose whether it activates or not, but only gets to choose at the time the rod activates. Disarm was deliberately reduced in power and effectiveness for this edition, so the Rod of Wonder doesn't get a disarm effect that happens with no save. That is too good to be true.
Are these weapons under the command of the caster or the target?
You are right, that one is very ambiguous.
You make any decisions for a spell cast by the rod unless otherwise indicated
The Dancing rune isn't actually a spell though.
But I would follow that pattern. The character that activated the rod gets to decide who the weapons attack.
Looking at the other possible results, there are a few that could be "situationally inconvenient" for the rod wielder, but none that are directly disastrous. So I believe the rod wielder decides who the weapons attack.