Mark Wilson has a levitation that would work in his Complete Course, but it requires a curtain. The typical way to levitate a child is a chair suspension, but the child needs to be at least four for it to work.
The appearance is easier. First stop is Mark Wilson's Complete Course where there is a cardboard box appearance. That requires choreography, so difficult to use with a one year old. There also is Mark's Tip Over Trunk.
There is a Paul Osbourne effect in the December 2009 Genii magazine called Santa's Helper. The child is produced from a box that looks like present (if I remember correctly ) I forget if there is any difficult choreography to that.
I think the best option is Jim Steinmeyer's X Marks the Spot in the December 2011 Genii. It could be easily made from a couple of pieces of wood and some foamboard. You would need to put it on a platform with wheels and bring it in - I don't think a one year old can remain hidden and quiet for a half hour while all the guests arrive. Better to pop them in "off stage" and have them "hide" for a couple of minutes.
Vanishes are difficult, in that they involve choreography.
Now that I've answered your question (so people can't complain that I don't give the answer people are looking for), I think using your one year old in the way you are suggesting isn't a good idea. First off, at that age they have limited ability to follow simple instructions. Second, I'm not sure that any appearance will work just for the amount of time that the would have to be "hiding."
If you post the ages of the other kids, I'd be glad to give you some ideas for effects.