One sheet of paper, small hands and a little patience: that's all you need. This is the airplane we use in all our cut-outs. It has a lock that keeps it folded, so it doesn't open in the air and glides beautifully. Here's how to fold it, with a drawing for every step.
I made it with my little dinosaur paws, so you can too! The most important thing: press every fold flat with your fingernail.

Fold the sheet in half lengthwise and open it again. That line is your guide.

Bring the two top corners to the center line. You get a roof.

Fold the big tip down until the roof looks like an envelope.

Bring the two top corners to the center line again. A little point peeks out underneath.

Fold that little point up over the corners. That's the trick that stops the plane from opening.

Fold the plane in half, with the little point on the outside.

Fold each wing down, one on each side, leaving a belly one finger wide.

Open the wings flat, straight like a table. Ready for takeoff!
Mark a start line with painter's tape and write down every record.
Put a towel on the floor a few steps away. Who can land on it?
Hold up a hoop (or a belt in a circle) and try to fly the plane through it.
From 4 with help, and from 6–7 almost on their own. Folding along lines is great fine-motor practice.
About 10 minutes the first time, and 3 minutes once they know it.
A symmetric one with firmly pressed folds that doesn't open. That's why ours has the lock in step 5.
Yes, as long as it's thin. Cardstock is too heavy.