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".
The word "i" can translate to "to/on/at" in sentences similar to those that you listed as examples. It also, however, can appear in Hawaiian sentences without an equivalent matching word in English. For example, "E hoʻihoʻi kākou i ke kī" -> Let's return (i) the key.
I guess one way around prepositional verb phrases is to replace them with a single word, hence, "return" vice "put back."
When I tap on the highlighted "E" in this sentence, it defines it (as well as ho'iho'i) as return. Why? So is "e" the verb or is "ho'iho'i" the verb? Does "e" actually mean "let", and are there any other uses for it other than being paired with "kāua/ kākou"? (Sorry so many questions!)
"E" can mean something like "let" when expressing a positive suggestion. For example, "E hele kākou/kāua" -> "Let us go" -> "Let's go".
It can also express a command, for example, "E hoʻomaʻemaʻe ʻoe i ka papahele." -> "(You,) clean the floor."
It can also express a sort of promise or intention of doing something, for example, "E hoʻihoʻi kākou i ke kī" could be interpreted as "We will/shall return the key." Another example: "E hoʻomaʻemaʻe ʻo Kaleo i ka papahele" -> "Kaleo will clean the floor".
So to answer your question, "hoʻihoʻi" -> "return/put back" could be called the "verb", but "e hoʻihoʻi" can be interpreted as "let (possibly us) return (something)", or is a command to "return/put back" something, depending on the other words in the sentence and other context.