You can use Sour Cream of Mayonnaise as a substitute of yogurt.
Your cream
yes
plain yogurt with a little vanilla, or sour cream with a little sugar and vanilla.
Yes, it should be just fine to do so. But you need to use 'regular' yogurt instead of the thicker types of yogurt, such as Greek yogurt. Otherwise, the filling for the pie will be too dense and dry.
Greek yogurt!
yes
yes
Yogurt.
Yes, but make sure it is plain yogurt, and that you mix it with a little milk to make it less dense.
Instead of eggs, some recipes use tofu, yogurt, or powdered milk.
I suppose you could, but the final product would be different than if you used oil. Because yogurt is acidic, you should add half a teaspoon of baking soda with the yogurt.
Yogurt is not a good substitute for butter in any recipe; yogurt is lower in fat, higher in liquid and far more acidic than butter. These qualities make it very different to butter, hence it's not a good substitute.
Plaiin yogurt can be a good substitute for creme fresh. If you're topping a dish with it you can definitely use it, if you're cooking it into the dish it may react differently so using it is a risk.