Wren, can you actually undo the tape on the mask base and show the flattened template please?
For everyone else, the mask gets its rounded shape from a dart cut in at the top. Just cut a straight line about an inch or so, fold one side over the other, and glue it in place. If you want an even rounder shape that will sit a bit higher off of your face (not quite glasses-friendly height, but maybe more sensory-friendly for some), you can make additional darts on the sides, roughly in line with the back corner of the eyes.
The cat muzzle uses a dart at the front to create a rounded shape. (Figuring out that one was a paaaaiiiiinnnn yesterday, but I did it!) All the others, you'll want to fold the center tab down first, then fold the left and right flaps over it. Play around with how tightly you fold them over to produce slightly different shapes, and glue it once you like what you see.
When looking to attach the different pieces, you'll want to cut lots of perpendicular flaps into the attachment points (highlighted in the picture below; you'll want to fold those parts up and glue them against the mask base). Having lots of separate attachment points will allow the cardboard to flex a little more and help with stability overall.
When fleecing or felting the base, you'll want to cut a piece that is.... way bigger than you think it needs to be. My technique is to lay the fabric on top of the mask, let it drape without pulling, and give myself about one inch above the tip of the furthest projecting part -- usually the ears. Cut a rectangle and start gluing at the bridge of the snout. Glue in sections, stretching the fabric to prevent wrinkles, until the entire piece is covered.
Then, and only then, do I cut around the shape of the mask, leaving about an inch margin around the whole piece. Cut flaps all around (like you can see in the second photo Wren posted) and glue them down, stretching as you go.
If you have any questions as you go about your making, please feel free to message me!