If I use Procession on Pillage, what happens? I'm guessing that I would gain 4 Spoils, but no action worth 6, since the Pillage was already trashed by the time Procession finished resolving?
This is the best I can find, which seems to be a definitive answer. The text in the Dark Ages rulebook is enlightening (page 7):
In rare circumstances an effect may try to move a card that is not where that effect expects the card to be. In those cases, the card does not move - the effect has "lost track" of the card. Losing track of a card prevents it from being moved, but does not stop anything else from happening. For example, if you Procession a Madman, Procession first puts Madman in play; then you resolve Madman, getting +2 Actions and drawing cards and returning Madman to the Madman pile; then Procession fails to put Madman into play again, because Procession expects to find Madman in play, but it is not there, it's int he Madman pile; then you resolve Madman again, only getting +2 actions this time, since it says "if you do" before the card-drawing, and you did not actually return it to the pile this time; then Process fails to trash Madman since Procession again expects to find Madman in play and it is not there; and then you can an Action costing (1) if you can. [...] Things lose track of a card if something moves it, if it is the top card of a deck and gets covered up, or if it is the top card of a discard pile and gets covered up.
This seems to say that the Pillage card goes into effect twice, even though it has been trashed at the end of the first time it goes into effect. So yes, you get 4 Spoils (assuming there are enough Spoils left in the supply).
From the explanatory text for Procession (page 16):
[...] then gain an Action card costing exactly more than it (even if somehow you failed to trash it). Gaining a card is not optional once you choose to play an Action card but will fail to happen if no card in the Supply costs the exact amount needed.
Which is to say, the text on the card doesn't say if you trash it, gain a card, only that you both a) trash the chosen card (if possible), and b) gain an action card costing one more than the chosen card (if possible). So yes, you do in fact gain an action card costing (6) in this case.