"he asks" is "il demande". "il pose une question" is closer: he is asking a question.
"He lays down", which is one answer given, is entirely incorrect. "Lay" is used transitively, ie when there is an object "He lays her down on the couch". Where the actions is intransitive, ie no object, "Lie" should be used. The correct answer here, where there is no object, is "He lies down."
Because pose is not about asking. It is not the english equivalent of "to propose". Check out the reference to the thread of things I have found out about the verb poser, also in this thread.
Its meaning is more about to put down something gently and with flare, with design. It is used in all sorts of medical and technical terms, and in common expressions. It is a great verb to learn about.
"he putS down" is like "il pose", it would need a direct object (his phone, for example).
here, I guess that in French as in English this is a second meaning of verb pose/poser, ie he is getting ready for a photographer to take a picture of him.
Not quite, I think. "drop" is better translated by "laisser tomber", which means that you hold something, then open your hand and the thing falls down. "Poser" is about quietly putting something on to a surface.
I have found out some FANTASTIC expressions, medical terminology and technical terminology that uses this very useful verb. Click here to check out some more information on the verb poser.
In these duolingo sentences it seems to be meaning " to put down gently, with purpose, with flare ".
I recommend you checking out all the other great ways you can use this verb !