I think these things can all come apart. None of these examples are uncontentious of course, but it begs way too many questions to assume that the three terms stand or fall together. The answer I prefer for this question is the generous one. Of course a person who believes that the world has no order, purpose or design can reason logically. We can reason reasonably about chaos, just as well as order.
Logic is the story about what follows from what in all situations.
Greg enjoys his job. That's independent of any order, purpose or design, as far as I can tell. But that's way too cheap. If you want to call that order then more power too you. But don't you dare attempt to infer from this that there's design or purpose or any of those more robust features.