I suggest two "general principles which lead to good questions in many concrete situations": making "abstract" results more explicit, and making explicit results more abstract.

I think these things are well-known to researchers, in particular I make no claim of originality or novelty; but, the question has been asked. And perhaps some student will read this. So, here is more detail.

I can't remember where I read/heard this, but there is a pretty well-known scale of explicitness in mathematical results, something like the following:

1. A non-constructive existence proof of $X$ (e.g., using a Noetherian hypothesis)
2. A "theoretically" constructive existence proof of $X$, or an "effective" theorem (in the sense of having some bounds or explicit information about $X$)
3. A method for constructing $X$
4. An explicit algorithm for $X$
5. A computer implementation of an algorithm for $X$
6. A usable computer program for $X$, meaning that it can be used by non-specialists

Of course not all of these are applicable in all situations, perhaps I left out some levels, perhaps some of the levels I included are a little silly... Anyway, the point is that with a scale like that, we can extract at least two "general principles" for generating questions in concrete situations:

Explicitization: Increase your position on the scale. If you have a nonconstructive existence proof, try to find a constructive one. If you have an algorithm, try implementing it. (It was a humbling experience to see that my beautiful theorem would take forever to actually compute a trivial case.)

Abstraction: Decrease your position on the scale (hopefully leading to generalizations or new ideas).

If I may use myself as an example, I've spotted a few proofs that used more or less explicit manipulations of big matrices, or coordinate charts, and I was able to simplify and generalize them by replacing matrices with linear transformations, coordinate charts with some basic topology, and so on. There's nothing special about my case. (And I'm not criticizing the earlier authors.)