It doesn’t mean thousands of litres per second to do it properly. A mask or helmet could be considered a poor tool for the job, though, because they are easier to fight/struggle with, and the person could hurt themselves in the process of that.
You need the normal level of air replacement for any given volume with a human in it, but you need to be using nitrogen as the source of air replacement. If you want to speed the process up, you could do 1x space volume/minute for a couple minutes, then drop it down to a normal rate of replacement.
Choosing a larger volume will not make it painful, but it will make it slower unless you increase the flow. But slower is not bad, per se, except that since it’s an execution, faster is possibly more merciful (depending on the person’s preference) because the person has less time to sit there and contemplate the fact of their death.
The suicide pods are pretty much the ideal balance of space taken. For an execution, perhaps a small room with a chair, and a somewhat faster nitrogen replacement rate (like, 60 volumes/h for the first two minutes, then 5 volumes/h after that).
Yes, precise circumstances matter.
For nitrogen asphyxiation in general, you do not have to be calm and cooperative for it to be painless, any more than you do so to stand in a room or sit in a chair.
As to the specific setup that the government chose, I can’t vouch for that.