This is the most general, casual term for the price you pay in a store.
あたい and 価値 refer to more general value, which isn’t necessarily correlated to a currency demanded in a store. The difference between あたい and 価値 is that あたい is of Japanese origin, and 価値 is of Chinese origin. They may be more or less common in various expressions or contexts. I think ね appears in news reports, at least I recall seeing it. Like 値が高騰している (a price is rising). I’ve never heard someone use it in conversation.
価格 is also the literal price of something in currency, like 値段, but it’s not a casual word. You wouldn’t use it in conversation at the store, but you might see it in store advertisements or articles about prices.
物価 is a word for the general prices in an area, like when talking about how it’s more expensive to live in Tokyo than a rural village.
For the second one, about global oil prices, I think 価格 would be appropriate, since you generally only discuss oil prices in relatively serious contexts.