Putting Podrick into play is part of the cost, not the effect (unlike Dolorous Edd, Ser Alliser Thorne, etc.). So if his ability is cancelled with Treachery, the save and kill effects do not resolve, but Podrick remains in play.
You can use Podrick's ability even if another copy of him is already in play. The card in your hand will enter play as a duplicate, the rest of the effect will resolve as normal.
If Podrick is killed upon entering play (e.g. due to Blood of the Dragon), it does not abort the resolution of his ability.
- Podrick can save a character from Trial by Combat, but not from The Seastone Chair. (Source)