We don't know the source of the vitamin D: For example it's entirely possible that the type of person who has higher vitamin D levels is the type of person who takes care of themselves through things like vitamin supplements, and people who take care of themselves are less likely to have preventable underlying conditions.
So, as always, it's important to remember that this study was retrospective, not experimental in nature.
If the type of person to take Vitamin D comes in for a test with less symptoms "just in case", then the probability that they are positive given that they are tested is lower.
No, if it's true, then it means that people taking taking vitamin D are more geared towards preventative measures, so they are more likely to get tested when in the absence of symptoms. More people who don't need to get tested would result in more negative test, so the inverse correlation is still the expects result.
Yes: There's evidence that lower-incomes are associated with less use of vitamin supplements [0]. And higher incomes would correlate to higher levels of health-care coverage that would allow for "just in case" testing.
So, as always, it's important to remember that this study was retrospective, not experimental in nature.