Multiple Advancement Questions

I've been working on some advancements for my mod, and have been unable to figure out how to make the following work as prerequisites:

1. Player kills Ender Dragon with specific item/items in inventory. I could link this to the vanilla advancement, but this either makes it so the player has to have completed both objectives separately, instead of at once; or it could cause an issue by making it so players who have completed the advancement already without the item cannot get the achievement.
2. Player kills mob by a sculk catalyst, triggering bloom, with a specific item in their inventory. Similar problems as above happen when trying to link it to existing advancement. 
3. Player receives a healing effect (either instant health or regen, whichever is easier to code) while a specific item is in the inventory. The original idea was to have a player throw a splash potion of healing/regen onto another entity with the item in their inventory, but I don't know if this is even possible. If it is, please let me know how I could do this!
4. Player has two specific items in their inventory at once. Like above, if I just make it without a custom procedure, it allows the player to accomplish each "objective" seperately. 

I've tried most simple solutions I can think of (I can specify what I tried for each advancement if anyone would like) but may have made a simple mistake in the code somewhere, I am pretty new to this and am probably making a lot of mistakes. I'm pretty sure everything I have made is just held together with duct tape and stings, but we all gotta start somewhere, I suppose.

Any advice appreciated. Thanks!