I'm assuming you mean valve gasket and not valley gasket lol I hate autocorrect as well

Head gasket can case coolant to mix with your oil. Easy way to check is if you see the coolant level go down while ur engine is running. I would normally test with water instead of anti freeze but don't know of you feel like draining and filling your system a few times in the cold lol

What your describing sounds a little different. Oil only burns like you described if it enters the combustion chamber. For it to both enter the combustion chamber and leak out of the engine.. And mix with coolant..R u positive the oil leaked from the engine and not the exhaust?

How fast are u loosing oil? U could have a busted piston ring. That would send oil into the combustion chamber very quickly.