Vayne rarely wants to build Shiv. This is because Shiv is a wave clearing item more then a DPS increasing item. Vayne is a duelist, and one of the strongest at it. He single target damage in insanely high. Extra mobility and the ability to move through minions really helps her out play opponents. I suppose...if your entire team has no wave clear at all then you might want it but her short range makes even that 1 AA on incoming waves dangerous
Akali often wants Abyssal "Scepter" because the early flat MR reduction works really well with her assassin kit. Remember with that and sorc boots you are taking 35 MR away which will often give you true/near true damage against enemy carries. Against Azir with his high area control having that MR also helps you survive entering his zone control. Abyssal Scepter is a great item for melee AP assassins because of the ability to tank a little extra and deal extra damage. However, if Azir is ahead by a lot 1 item isn't going to make a huge difference. When he gets too far ahead look to roam and gank enemy lanes to catch up and just catch your waves when you can.
Any other item questions?