My concern about the wood is wouldn't it be easily breakable? Like, if it's balsa wood or something? Cardboard can bend without breaking (for the most part), not so much with balsa wood.
And Slackbot, I've had similar problems with gasket rubber myself... one common problem I've run into is that for some reason, the foam rubber likes to pull on the rubber, especially the jaw, so it results in the puppets looking like they have double-sided sneers. It kind of worked well with an elf puppet I built, because the character was meant to be sort of sassy and sarcastic, but not so much for others.