How to check for a armor id and or display name?

To my knowledge, there is no way to find the ID of an item on an entity. If you want to find if they have armor on, or how much armor value an entity has, use the "Armor value of provided entity", located in the "Entity procedures" tab. Good luck on your searches!

Why do you need to type it in? What's wrong with selecting the item?

Dependencies allow you to do that... Though I don't know how to do that just yet myself as I'm soon going to test it out.

Well, if you really want to, you can get the ResourceLocation suing Item.REGISTRY.getNameForObject(item), convert the ResourceLocation to string and then compare the string to whatever you want, like this:

Item item = stack.getItem();
if(Item.REGISTRY.getNameForObject(item).toString().equals("minecraft:diamond_helmet")){
//do stuff here
}