Although this is my own experience, it might give an answer to your first question "if something is obvious, then it is obvious that it is obvious (so why include it at all?)." There are times when I will write, "clearly," or "of course," when I feel leaving them out might insult the reader's intelligence. If we imagine your example
If d divides a and d divides b, then of course d also divides a+b,
(probably without the italics) in an advanced context, that is, as a prelude to invoking an important theorem or stating something not so obvious that depends on that simple fact, then I might look at it that way.