Does ability power affect nonstandard abilities? If so, how does it affect them? Examples: Shapeshifters (Does it affect post-shift autoattack damage?) Graves, Kass, Vayne, and other manaless champs I may have missed (Does it affect AA splash damage, mana burn/shield, and % hp damage, respectively?)
Any info about how those interactions work would be greatly appreciated!
I really want to know this too! Where are the knowledgeable bunch you normally find on here
Ability power only affects abilities that have an AP ratio. Not every ability does.
Some champions have an AP ratio or two tucked into their kit somewhere, like Jhin Q and E, Graves W, and most recently, Qiyana's W, but the ratio isn't there to encourage to you build AP. It's to make Baron buff more potent on those champions and to maybe give them the option of building an item that has stats/actives that they want, but would otherwise waste the AP, like Rageblade or Zhonya's.
If you want to see whether or not your champion has AP ratios, check the tooltip in-game or go to the wiki. Green numbers usually indicate an AP ratio.
