The first thing to do is make sure you are taking advantage of automation as much as you can. Your email can be sorted automatically into different folders, allowing you to choose which ones you don't need to look at now. Mailing list subscriptions, for example. You can also automate, at least partially, how things move from your email inbox to your action lists if you're using the right toolset.
You can also reduce friction on your voicemail processing if you use a service that does voice-to-text and sends you email. The transcriptions are never perfect, but they're often good enough to understand, and I read even bad transcriptions a lot faster than most people talk, saving me the listening time. Google Voice does this kind of thing, there are other services that do too.
Overall, you have to get through your processing faster. You aren't going to reduce the number of items that hit your inbox (well, unless you change jobs or have a big failure that reduces your responsibilities, which I suspect is not an acceptable solution). You aren't going to have a lot of success changing other people's behavior, although you may be able to get them to send you an email followup after they appear at your desk which could be a small help. So you're left with getting through it all more quickly.
Be careful to limit how much work you do as "processing". Don't let "organize", "review" or "do" sneak in to processing time. An inbox item that triggers a new project might best be entered into your system as a "Plan new project" next action, rather than being added to the project list and an action created right then. For another example, I have a lot of email that gets processed from my inbox to a @Reply folder, with a Next Action on my task list to "R&R [email subject]", meaning Research and Reply. That processing takes under 2 seconds, with an Outlook Quick task and keyboard shortcut. If it's a really short reply, (2 Minute Rule) you can do it while processing, but be certain you are really staying under 2 minutes. I periodically set up a timer and reset for each email to check myself. Yes, it slows down the processing for that session, but it is good to retrain my sense of how long (or short!) 2 minutes really is.
If you have people working for you, or colleagues who share responsibilities, hand off categories of work. That can let you get whole chunks of communication out of your inbox. Simplify tracking that work by making reporting on it part of the delegation, don't plan to go ask about it.
Look for clues that help you discard things faster. "Delete" is a valid outcome from processing - not all good ideas need to be done. And not all ideas in the inbox are good. There may be classes of things that hit your inbox that never need action from you, in which case you can automate (step 1) getting rid of them.