After 45 Minutes of talking to different but none the less helpfull people on the Hotline I was asked to change my Apple-ID password.
This fixed that one part of not being able to send images to other iMessage-Users.
The other thing was rather stupid...the person I've tried to send pictures to had a full storage. His iPhone was simply incapable of saving any more images. So keep in mind, that the problem might not be at your end...
Thanks benboyd5 for this resource, it was really helpful. It did contain some suggestions already given in this thread, but it also gave some others, and gave them in a progression that helped me a lot with testing. For me, a helpful clue was testing iMessage by turning it off on all iOS devices, rebooting, then trying turning it back on, on one device at a time. It ended up working on both the iPad and iPhone at the same time. I also tried almost all the other steps as well. Thanks! I'm in a much better frame of mind now that it's working again!