Learn Russian: Он ни у кого не спрашивал, что положить внутрь.

положить is a transitive verb of action, meaning "to put" or "to lay" something down. You use it when someone is actively moving an object into a position. лежать is an intransitive verb of state, meaning "to be lying down" (describing where something already is). Since he is figuring out what to actively put inside, we need the action verb.

Could we use класть instead of положить?

You could, but it changes the aspect. класть is the imperfective pair to the perfective положить. Using положить focuses on the specific, completed result of putting things inside on this occasion. Using класть would imply a repeated or ongoing action, like asking what he generally ought to be putting inside.

Why is the preposition у sandwiched in the middle of ни у кого?

Whenever a negative pronoun like никто (nobody) or ничто (nothing) is used with a preposition, that preposition always steps right into the middle, splitting the negative word in two. The preposition у is needed here because the Russian phrase for "to ask someone" is спрашивать у + Genitive case (кого).

How does что положить translate to "what to put"?