First of all, who is they? It seems like this is an individual woman. That's part of the appeal, that this isn't a mass market product, but a work of love made by a real person for a real person.
The uniqueness of the toy is part of the appeal as well. If other people order the same toy, first of all, what is the point? You can buy stuffed toys anywhere. What you can't do anywhere is order custom stuffed toys. But more importantly, what does duplication do to the appeal of the original toy?
Adding competition would be equally disastrous I think. It makes the whole process a game, and anything that can be gamed will be gamed. If there is any reward at all in having the most popular toy, even just recognition, then you can bet that adults will start making drawings and saying they were from children. Further, the competition will distract the children who draw from focusing on the specialness of having their drawing come to life and instead turn their focus to how well their drawing is performing.
Your ideas may make a popular and profitable business, but it would be fundamentally different from the business here. You can't scale personal connection, and you can't duplicate uniqueness.
Don't get me wrong, I think it's great that you thought up all these ideas. I just don't think this woman should apply them. What she has is special, profitable (probably even enough so to live on), and makes a positive impact on the lives of her customers.