Wow, you really know how to analyze English very well!
Let me try to answer to your questions.
However, I am not a grammar or linguistics expert. There may be someone on the forums who can answer your questions better than I can.
I think that "Dad gave a car to me" is not ditransitive because there is actually only one object - the car.
(I may be wrong though).
I think that in "Jane took off her coat" the verb is transitive because "her coat" is the object of the phrasal verb "took off".
My guess is that in "asking for information", it is intransitive because the "asking" is not an action that affects the "information" directly.