I don't think you can save the iMessage (you could try accessing it from iCloud.com on a computer and seeing if there was a way to save them) but if it's just the pictures you want to save you can go in individually and save each photo to your camera roll...then sync the pics to your computer. A bit tedious but it's an option
If you back up to iCloud (or iTunes on a computer) and restore a new iPhone from that backup, you'll get your iMessages back. If you move to Android or some other operating system, those iMessages are not recoverable on your new phone. They'll still be in that backup file though.
I think they want to actually be able to access these photos on their computer and not just have the iCloud backup which you can't view files on unless you put it back on an iDevice.
You need an external program that installs on your PC or Mac. I have never used these, so I cannot vouch for them. Consider something like copytrans.net Copytrans Contacts which saves the conversation to a Word document.
This is true. There are a lot of free programs to rip your messages/pictures and save them in many ways. I use one