I created this cute little laughing face animation by drawing the individual faces on my iPad and using an online collage maker to put them into a texture sheet 'format' and applied the Texture Sheet Animation module and setting the emission over time to 1. At first, it wiggled around quite a lot, but I fixed that issue by changing the Start Speed to 0. Now it is much better, but it still seems to wiggle a little bit in place.
I know that the faces are lined up exactly, because I simply duplicated the layers as I changed them in my drawing program (ArtStudio for iPad), not changing the outside circle at all, and I used a perfectly square grid layout when I made the collage I am using as the texture sheet (using BeFunky.com). Is this slight wiggle just unavoidable when using a particle system as an animation?
It's probably the particles emitting at slightly random positions. This lesson goes over how I created a sprite based particle effect: https://cgcookie.com/lesson/creating-an-explosion-particle-system
If you remove the texture sheet and just use something like a square texture you should be able to see if the particles are emitting perfectly in the same spot.
I was trying to follow along with the video, but this option doesn't appear on the create menu when I right click
It's hard to see from the image, but I believe you're trying to create an image sequence. You'll need to download the tool in order for that option to be available. That tool just creates the sprite sheet, since you already have one made you can skip that part. Look at the video where I start creating the particle system at 6:15 onward. I go over how to use the sprite sheet with a particle system.