A moderator initially sent a private message on Sept. 23 2014 to me, to which I then responded. I still haven't heard back and genuinely need a response, so what can I do? The system prevents me from writing again until a moderator replies.
Moderator messages are not really meant for discussions, in most cases there is only one message from a moderator, and maybe a reply from the user, and not more. The ability to reply is meant for clarifications, if necessary.
Moderators are not obliged to reply to any messages there, in many cases it would not be productive to do that. Such discussions e.g. about suspensions tend to get nowhere, or they escalate to insults. Discussing every suspension would simply waste moderator time.
This doesn't mean that moderators are ignoring users, or don't have to explain their actions. They must explain the issue and their actions, but that happens in the first moderator message. They are not required to engage in a back-and-forth discussion, it is enough to state their case once.
Some issues also belong in other places, if you want to e.g. challenge a specific rule this is about, it would belong on meta where the community can weigh in. Contacting Stack Exchange is also an option.