You can have multiple h2 elements on any page (and there is no law against having multiple h1s, though it's not best practice). You can also have different h1 and h2 elements on different pages; h2.centerBoxHeading is a separate item from h2.greeting.
Since selectors with equal priority are ruled by the last occurrence in the stylesheet, best practice is to put general sitewide selectors first, followed by more specific ones and exceptions to the general rules.
h1.indexDefaultHeading will do nothing, as there is no class named indexDefaultHeading. #indexDefaultHeading is the way to indicate the name of an id.