Correct. It is a directional object marker, so in this case the key is receiving the action of the command "to return."
Yep! E (verb) kākou... == Let's (verb), which then can add on the object the verb is directed towards, e.g. E hele kākou i ke kahakai. == Let's go to the beach.
I agree. "To put back" and "to return (an object)" seem to have the same English meaning in my opinion.
"I" is a directional marker. It points out the object that receives the action. So, "E hoʻihoʻi kākou" (Let's return) "i ke kī" (the key). In ʻŌlelo Hawaiʻi, almost every time you act upon an object, that object should be preceeded by "i".
I guess one way around prepositional verb phrases is to replace them with a single word, hence, "return" vice "put back."