If you live in the US, then Google Voice might do what you want...except for that part of using your existing iPhone number (with the exception of if you use Sprint...Sprint has a deal where you can use your Sprint number for your Google Voice number...for others, you would get a new number).
I use Google Voice as a way to keep my old cell number active (and not have to update everyone with my new number).
Technically, Google Voice does not support MMS, but I get text messages with pictures (aka MMS) by way of the Google email address that is associated with my Google Voice account when someone sends me a text message with pictures.
There is an iPhone app...not sure if there is a Chromebook app (looks like there might be something in the Chrome store), but you can certainly log into your Google Voice account from a Chromebook.
I will note that when you get a message by way of Google Voice, it also comes to the Google email address that is associated with that Google Voice account. You can then reply to that message using email as well.
Note that if you do have Sprint and activate Google Voice on that number, then Google Voice will "take over" for your voicemail service. You can read more here: https://support.google.com/voice/answer/1207070?hl=en