The common phrase "your take on X" means your opinion about X, or your interpretation of X or your belief about X. It can mean any of these related things.
The phrase "Your takeaway from Y" means the conclusion you draw from the event or set of events Y, or possibly the moral or lesson that you derive from Y.
In these expressions "Take" pretty much always uses "from" as a preposition, and is normally used for a conclusion or belief about or analysis of X. "takeaway" normally uses the preposition "from" and refers to some sort of summary or result of Y.
In many cases either phrase can be used, although some rewriting or rephrasing may be needed to keep the resulting sentence natural.
In the question's first scenario either could be used, but I think "take" is more natural.
In the second sacenario, one could write "what do you takeaway from this matter?" or, better, "what is your takeaway from this matter?" orm in my view still better, "what is your take on this matter?"
The expression "your take on" uses "take" as a noun. One could say "what do you take from that" using take as a verb, but I think that would be less common. "takeaway" in this construction is always a noun. When take is used as a noun here it is more natural to use it with "on" not "from". "takeaway" should, I think, always be used with "from". They are different words, with different although related meanings.
Adapted from responses to comments:
The forms "what do you take on his behavior" or "what do you take on this matter" would both seem unnatural to the point of being simply wrong to me. Both seem to be using a verb form of "take" in a construction that calls for a noun.
The expression "your take on" uses "take" as a noun. One could say "what do you take from that" using take as a verb, but I think that would be less common. "takeaway" in this construction is always a noun. When take is used as a noun here it is more natural to use it with "on" not "from". "takeaway" should, I think, always be used with "from". They are different words, with different although related meanings
I would not say: "what do you take away from this matter" or in general "what do you take from X" to mean "what is your conclusion about X" or "what is your onion about X". I can't quite say it is wrong, but it seems to me quite unnatural in US English and I think in UK English as well. I would think that a non-native speaker had misunderstood the noun "takeaway" which is used in a metaphorical sense in such a construction. (I am not sure if this is natural in other varieties of English such as the Indian.)
