You can get most of that if you buy into the Google ecosystem.
I use Gmail for all of my email. I have several addresses. Those non-Gmail addresses I have are auto-forwarded (or pulled via POP3) into my Gmail inbox. I have Gmail set up to send as any of those addresses.
I use Google Voice and only ever give out that number. Any SMS messages sent to my number are forwarded to my Gmail. I can respond to those message by replying to the email message.
Voicemail, similar, is sent to my Gmail. While I can't listen to the voice message right in the email client on Android, it's just a hop and a skip over to the Google Voice app. In any case, I usually just use the automatic transcription to gauge whether I want to call the person back or not.