For your interest greasy is used in England. In fact, a cafe in England that serves 'English Breakfast' is often called a 'Greasy Spoon'. A greasy spoon serves fried food. You tend to get chips, eggs, beans, sausage and bacon. It's much like the components of a full English breakfast but eaten throughout the day

"Greasy". It is always negative. If there is too much oil, I would say "there is too much oil". I would never say that it's "oily". Saying that would make it sound as if I were talking about an automobile engine.

I would rarely use them interchangeably. As foods, oils are liquids that come from plants whereas grease is solid at room temperature and comes from animal fat. ‘Greasy’ has a more negative connotation by default.

In the UK both are used.

In the US, they're more or less equivalent.