I think most of these questions are fine, because new answers can be added when more information is available (or better: existing answers edited). I think we need to rely on the community to clean these up as much as possible.
The problem with trying to restrict this is that nearly anything could be an ongoing series, since trilogies get second trilogies or become series, and recons are so common.
Other SE sites have answers that get out of date as well (e.g. when a new version of a programming language is related, or when the problem is due to a bug that gets fixed).
For the "In today's episode of X, we see event. What influence will this have on character?" case, where the answer is going to (or very likely to) be answered in a week or some other short period of time, we are becoming a reference site for X - in the future when people watch episode A and have the same question, they'll find us via a search engine and get the answer (hopefully edited and improved as more is revealed).
However, I don't think prediction questions are useful (but I would downvote/not vote, not vote to close, in most cases). i.e. "In today's episode of X, we see event. What do you think will happen as a result?" These are discussion questions, so are inappropriate in general. "Speculation" questions are considered on-topic, but I believe this generally referred to matters outside of ongoing storylines.
Who is Dr River Song, and what is her relationship to the Doctor?
Although this cannot be definitively answered yet (unless Moffat joins and starts giving spoilers), what information is available can be given, along with "this is being explained over seasons 5 and 6". Eventually, the best answer will contain a complete and definitive explanation (i.e. once the show has given all the data). Our site remains a good resource during the whole lifetime of the question.
We found out in episode X that Dr River Song is imprisoned. What's the most likely reason for this?
The question encourages bad answers - i.e. ones that are purely discussion or "wild ass guesses".
Why was Dr River Song imprisoned?
Correct answer now: "we don't know yet, this is being told through Season 6. What we do know is ...". Edited and correct answer later, "She ...".