If I find that while answering one question I actually end up answering another, is it okay to post my answer under both questions? Or is that in poor taste?
Posting identical answers to different questions is a bad idea: if exactly the same answer solves both problems, then the questions are likely to be duplicates. (In fact, it's considered such 'suspicious' behaviour that doing so automatically raises a moderator flag.)
The above are my own thoughts on the matter, which seem to align with the policy laid out on main meta:
Copying and pasting the same answer for different questions is not the correct thing to do.
If the exact same answer is required for different questions, then those questions could be duplicates; it could also be the user is generalizing too much the question being asked, and writes the same answer for different question.
In the first case, it is more appropriate to vote to close the question as duplicate; in the second case, it is more appropriate to reformulate the answer, and make it more specific for the question being answered. Even if the solution to two different questions is the same, copying and pasting the answer given to one of them is not what I would do, as there is always a part that is specific for a question.
While it doesn't relate to your set of questions, I will note that the various "Identification" questions lend themselves to this as sometimes you'll have multiple questions that have the same details that match your answer but could still be distinct works (and, for that matter, the system doesn't allow closing them as duplicates because they have no accepted answer).
That said, usually there are some additional details that differ that you can elaborate on such that you're giving the same ultimate answer, but explaining different parts of it.