Not really. Appointed means that Jane will definitely get the prize. Nominated means that she is one of two or more who will be considered for the prize. This suggests a nominator. Shortlisted means that there has been a process of elimination, but Jane is still being considered. For example, "Eight people have been nominated, and Jane is one of three who have been shortlisted." I hope that's clear. Someone else may be able to explain it better.
If I'm being considered for an award or a job, I'd rather be "shortlisted" than "nominated." The former suggests that I am viewed favorably. Of course, there could be three people in the list of nominees in one context but five people in the "shortlist" in a different context.
Another difference is that a nomination is often formally recognized, while being shortlisted is usually unofficial.
These words are definitely not interchangeable. They do not have a shared meaning.
Other people nominate you (put your name forward) to receive a prize or award that is available. From the list of nominated people, the judges prepare a 'short list'. if you are 'shortlisted' you might still win the prize or award. Yes, even if you don't end up winning it is a better honour to be shortlisted than just nominated - you got nearer the prize.
One cannot be appointed to win a prize. You can be appointed to a position or to a job or to perform a function. There is no competition involved in this case.
I suppose it depends on what you mean by "official." Unless I see a press release or official website with the phrase "shortlisted" (or a close variant), I might not call it "official." Again, it depends on the situation.