Q: If I play a Shen Gong Wu card into my Vault or I play it as a Boost, and it says, “Draw a card,” do I get to draw?
A: No. You use the special ability text on a Shen Gong Wu only when it’s in your Vault and you tap it to help you in a Showdown. If you put a Shen Gong Wu into your Vault, its ability won’t do anything right then. If you play it as a Boost, you just add its Power. If you put it in the Showdown Arena so you can fight for it, its ability and its Power have no effect.
Q: Can I play a Shen Gong Wu directly into my Vault and challenge my opponent to a Showdown?
A: No.
Q: If I flip a card off the top of my deck to start a Showdown, but it turns out not to be a Shen Gong Wu (so there’s no Showdown), can I now decide to play a Shen Gong Wu from my hand to start a Showdown? Or can I decide to play a Shen Gong Wu directly into my Vault now?
A: No, you can’t do either of those things. Your turn is over.