I understand 凭 as a verb to mean rely. But I often read it and the seemingly related 凭着 in familiar contexts but still do not know what function it serves. Seeing example sentences doesn't make it any clearer. In what cases is it most used?
You are close. 凭着 expresses the idea of based on, according to or relying on, and when translating into English, prepositions such as on or by can often be used to convey this. You should be able to use it in situations where this concept needs to be expressed. As you suspected, its part of speech is best thought of as prepositional. 凭, when used in this sense, is equivalent to 凭着.
Keep in mind that translation is an art and you do not need to stick with the words that I used. Be flexible.
It means "use". Usually the thing that is used has some specially importance, that is, if this thing is not used, people will not complete the task successfully. For example: