The reason you're getting dairs is because you're pressing down when you press A/Z. Here's the frame breakdown of how float nair goes:
KneeBend - 5 frames
Jump - 1 frame
Float - 1 frame
Nair - 3 frames (4 if you hit something)
Landing lag - 4 frames
You need to be pressing down on the Jump frame to float at the next frame. You need to be pressing A on the Float frame to nair at the next frame. When I float nair, I jump and press down at about the same time (it's a bad habit, I spotdodge OoS a lot when I want to float because I press down too soon). You can press down during any of all those jump frames and hold it through the Jump frame. You have to make sure you let go of it by the Float frame. In reality, that's very hard. You'll probably end up getting a few extra frames of float where you're still holding down because we're only human. Those extra frames of float will hardly matter in any situations.
What matters a lot is fast falling the nair. To do this "optimally", you need to input fast fall on the first frame of nair. What this means is holding down on the Jump frame in order to float, then letting go for one frame to nair, and then pressing down on the next frame to fast fall. THAT's the hard part, that sequence right there. I'm not even sure control sticks move up to neutral position in one frame. In reality, it's rarely ever necessary to do that. You'll notice in the first gif you linked that Armada has to float forward a tiny bit before every nair. This increases the number of frames spent in float and lengthens the time between down inputs making it much more reasonable, but still hard.
Another thing Armada does that makes this technique easier is that, when he hits someone, he fast falls after hitting them. The places numerous hitlag frames between the float and the fast fall input making it very easy to just float foward, let go of down, nair, and fast fall later. However, this method makes Peach lose a few frames of advantage on shields.
One more thing, letting go of float as you start the nair is pretty important in trying to land as soon as possible. You can't fast fall while in float, so if you float too long, the float can eat your fast fall input.
A final note, Sheik could have ASDI'd down any of those nairs and shielded before the next one happened (well maybe not the later ones because he DI'd up popping him up into the air). Honestly, these quick nairs are not very useful and amount to nothing more than a neat party trick. FC fair to FC nair or nair to nair can sometimes be useful, but those situations are mixups and all the top tiers have ways to deal with it.
