What's the difference between alchemizing with AND (double punch a card) versus OR (overlap two cards)? They both combine two items, but is there a consistent rule for what aspects of which things get combined in which way?

No consistent rule, but the more you && combine a thing, the closer to 00000000 (Perfectly Generic Object) the card will be, and the more you || combine something, the closer to !!!!!!!! (Perfectly Unique Object?) it will be. As such, when combining multiple items, you have to take into account that the more you abuse && or ||, the less variety of items will show up, just because more or less holes will be punched on the cards.