'Would/do you mind' + whether (polite indirect questions)
Although both if and whether can elicit binary answers, if strongly references one option (the stated option) whereas whether references both.
The proposition "I sit next to you" has an implicit negation: "I don't sit next to you". The question "Do you mind?" with reference to this proposition could have 4 possibilities:
- I mind that you sit next to me.
- I mind that you don't sit next to me.
- I don't mind that you sit next to me.
- I don't mind that you don't sit next to me.
Asking "Do you mind if I sit next to you?" references only the proposition and not its negation. Hence a simple yes/no answer distinguishes possibilities 1 and 3 above.
Changing if to whether changes the question. A "no" now means that they don't care whether you sit next to them - i.e. they don't care if you sit next to them, or if you don't sit next to them (possibilities 3 and 4 above). A "yes" is more problematic because it picks up both 1 and 2 - leaving you with no way to satisfy them. This might be the source of discomfort with asking the question using whether instead of if.
The train example doesn't have this problem. Although the proposition "the train has left" also has the implicit negation "the train has not left", the question "Could you tell me" is more of a request than a true question.
If it were a true question, then "Yes" would mean "Yes, I can tell you" rather than "Yes, the train has left", and likewise with "No". In practice, people would tend to recognise the request to convey the train's state, and reply with something like "It's left" or "It's still here". That is, you only have two possibilities to consider, not the four that accompanied the earlier example.
In this case (with the train example), both "if" and "whether" produce similar answers because a negative answer to an if question is equivalent to a positive answer to its negation. However, compared to a whether question, the if question still emphasises the asker's interest in the positive case.
More generally, you can think about whether questions as if they have an implicit "or not" tacked on, whereas if questions don't.