Hello, simple but essential question:

Can I trigger an ability from a card that comes out of shadow as it comes out of shadow? For example, can Tyrion trigger his own ability as he comes out of shadow?

thanks

The anatomy of the thing:

1. Initiate bringing the card out of Shadows (pay gold, etc.)
2. Save/Cancel bringing the card out of Shadows
3. Resolve card coming out of Shadows: the card is FULLY "in play" at this point.
4. Passive effects activated by the card coming out of Shadows
5. Response effects to the card coming out of Shadows

Now, since the card is fully in play in #3, and you act on appropriate triggers in #4 or #5 (depending on if it is a passive like Varys or a triggered Response like Tyrion), any text on the card that just came out of Shadows is fully in play, active and able to be used. So yes, Tyrion can Respond to himself coming out of Shadows.

In the end, this is no different than something like Dragon Thief triggering it's own "after this card comes into play from your hand" Response immediately after coming into play.