This is a very interesting question for a thread.
I can only speak from my personal experience however. For a musician, it is indeed expected to do extensive research.. it goes further than just the notes and markings on the page, the context or the origins, there is study of style, performance practice, musical analysis - formal and harmonic,etc. Composers rarely divulge directly and /or fully their intentions, and if one could ask a living composer about their work and they deigned to answer, well the ones that are dead, we will never know. Of course, for an opera, the libretto is revealing but for what we call "pure" music (by opposition to programmatic music) a lot of work needs to be done and the interpretation can vary immensely from one performer to another. A lot of the titles given to pieces did not come from the composers themselves but from editors/ publishers or from performers/the public. A good example, many of the Chopin Etudes have nicknames, none of which were given by Chopin himself. However, for figure skaters, I do not think that it should be expected that they have such extensive musical knowledge. In other words, the work to be done by a performer (musician) goes quite deeply...Some analysis I have written are over 100 pages and cover many areas of research. So I cannot imagine that an athlete would have the time, nor the knowledge, nor the formal training, nor the interest to go that deeply into a piece of music. I also don't believe it's necessary for figure skaters.
So for figure skaters, my answers are all very open... no absolute answer either way, and if sometimes, I have strong reactions, even objections to some choices, music is NOT the main part figure skating program but a vehicle to allow for displaying skills and compete... So, I have learned how to detach myself a bit more... and after a strong initial reaction, I will let things go
I tend to be much more critical about bad music cuts than skaters not characterizing properly their chosen music.
Here are some reasons on top of my head.
1) Not all skaters have a loud voice in choosing their programs, especially younger ones. Coaches and choreos will suggest things and that's it... For instance, how many young girls would really like to skate to Méditation de Thaïs or Big Spender if they really knew what is going on in there??? But, if given by their choreo or coach, and if they like the music or even if they made an uninformed choice just based on how things sound like, can we really blame them?
2)I think that with many skaters, there is a general idea about what is happening in a piece when doing programmatic music... Inspiration is drawn from the music and plot of the opera for instance...but the skaters will not portray line for line what happens... which is very fine with me. Let's be thankful that all these Carmens or Mimis or Toscas or Juliettes do not all die on the ice

So, I don't fault them again for not going with the storyline. So that would answer the last question in the OP... Skaters do it all the time, not depicting what truly happens... and that's very alright especially considering that a lot of the music used is a testimony of cultural concepts that no longer would be accepted nowadays... (if you have not clued in by now, 19th Century opera is super misogynistic, probably just like society was back then )
3) When skaters make a case of trying to tell a story, and they reveal that this is their approach and focus for the program, then, yes.. they should really do some research, at least about the story and context. Some do. Some do not. In the end, can we really fault them ? I don't think so... I do appreciate when there are attempts to stay "true" to the work, even if the attempts are not so successful.
In the end, I mostly get mad at coaches and choreos for choosing inappropriate topics (in my opinion) for young skaters... but I do not expect skaters to know so much about the music itself and its context.