a) Who did you approve of t studying linguistics?
What does the t stand for?
Hello again, Navi—The italicized "t" stands for "trace." It can also be represented as a "gap," as in "Who did you approve of __ studying linguistics?" Essentially, the trace or gap represents the place from which the interrogative pronoun "moved" as the sentence transformed from its deep structure (roughly: "[+Q, +past] you approve of whom studying linguistics"] to its surface structure ("Who[m] did you approve of studying linguistics?").
Are you sure that sentence isn't supposed to mean:
b) Whose studying linguistics did you approve of?
You approved of him studying linguistics.
That's exactly what it's supposed to mean. It's just that, instead of the POSS-ing (possessive-ing) construction, which you have in (b), we have the ACC-ing (accusative-ing) construction. I am not sure whether (b) is grammatical. I tend to doubt it. The following, in which the preposition is fronted, too, seems closer to being grammatical, though even it may not be fully so:
b') Of whose studying linguistics did you approve?
Similarly, we could revise your ACC-ing specimens in your opening post to the POSS-ing construction and use Pied Piping on them. In each case, we have a fringe construction of extremely dubious grammaticality, it seems to me:
1') The accident about whose happening I was worried finally happened.
2') The expensive plate about whose breaking I was worried broke.
3') The accident of whose happening I was afraid finally happened.
4') The expensive plate of whose breaking I was afraid broke.