I would use the jugs of liquid starch.  I think it holds better than the average flour/water mix.  You might even want to dilute it with water, maybe down to a raw-egg-white consistency.  The thicker the starch, the longer it will take to dry.

I would have the kids apply 2 or 3 layers of paper to the balloon first.

I would make the cones out of construction paper, if possible.  Make them longer than you want, and cut 4 slits around the base and fold those flaps outward, then glue the flaps onto the existing layers of PM, anchoring with more strips of PM.  When you're making the cones, make the first one the size you want, then roll the others around that one, to make them all the same size.  To keep the cones from collapsing when moist, stuff them with crumpled newspaper to keep them solid.

I know that the popular method of applying paper strips to a form says to immerse the strips in the adhesive, but getting them that soggy really doesn't work too well, esp with the time restrictions you've got.  If you have access to brushes, I would suggest that you  have each kid work on a thick mat of newspaper, and brush the starch just onto one side of the paper, and not too heavily.  If you don't have brushes, they could run the goopy strips between their fingers (scissorslike) to get rid of the excess starch.

Good luck, and have fun!

Sue