Famtasia was used. No warps or passwords used.
In NG3, as in other NGs, the sword slashing animation causes you to stop, unless you are in midair, and even then if you land during it, it must finish before you can move on. You can interrupt this animation by jumping again immediately, I do this quite a bit. There is one static scrolling stage, I show a few holes in the scenery there by entering walls, going through a couple platforms etc, but its more for fun than speed. Other than that I don't think there were any errors really abused. A few obstacles are avoided while invincible after a hit. Certainly I make use of the windmill star a little more extensively than was intended, but that's not quite an error.
I only take a hit when it is faster than waiting or killing the monster. When I do wait it is either to avoid a hit which would propel me backwards or it is for a game event like the spikes in the final stage. If i wait I probably tried other ways and found waiting was the only way. The spikes for example do a huge amount of damage so it was only possible to go through them once.
The bosses only take one damage per hit no matter what you hit them with so i generally use the sword since it fires the fastest. The exception would be the special weapon that fires up and down; since there are two projectiles it can do twice the damage potentially. It might be possible to lower total boss time by a couple seconds by using this. Likewise small optimizations may subtract a few seconds from the time but I am unaware of any exploits or shortcuts which I did not use.

PS, Famtasia draws the jungle stage totally wrong, I had to play it the way it is with invisible platforms and all. It was pretty weird.