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Game Over screens and the quick-save code both increase the KO counter (by only 1) each time they occur. As stated, losing lives does not increase the KO counter by any degree.
I found it odd that even the quick-save code would count against the player, because this wasn't the case for the first Legend of Zelda. So it's quite impossible(*) to have a 000 file unless you play through the entire game in one sitting, which will overwrite your progress and start you on the 'second quest' with your old spells and levels intact.
(* Although in theory, the KO counter's value might roll over to 000 if you Game Over / quick-save 256 times)
The save/continue screen increases the counter regardless of how you made the screen appear (either by losing your lives or with the code), yes. While we're talking about the second quest, or New Game+ which is probably more accurate, I may as well note that it resets the save counter to zero once you start it.
Can't quite remember offhand if this counter rolls in Zelda II because I might be getting my Zelda games mixed up. If it's like LttP at all then it would max out without rolling.