I have an iPhone,
There is no app that will allow you to record incoming call, Apple will not allow this. There are a couple of apps that will record outgoing calls.
For Voice mails I had to connect my phone to my computer with a 1/8" male to male audio jack and then play each voice mail and record it using some type of audio software on my computer, there is no way to move the audio files from the phone directly.
For texts there are a couple of programs for your computer that can retrive text converstiaons from the iPhone back up on your computer. These may now work for voice mail as well, but I'm not sure.
Now you can also jail break your iPhone and I think there are apps that will do all of the above directly, but jail breaking voids your warranty.
Apple is still fairly strict on what they allow app develops to do through the app store.
I think the android apps have more flexibility.
"Reality is that which, when you stop believing in it, doesn't go away"
Philip K < edited >