So, I got this Lumia 1020 for testing.
And the first problem i have to solve is how to get my data to the new phone (contacts, emails etc)
Nokia has foreseen that a switcher from iOS to Windows Phone will probably get his/her contacts. So they have built an app for that - that is also preinstalled:
It's super-easy to move your contacts (and also photos if you wish) from your "old" phone to the new.
2. Receiving SMS
For phone functionality i just put in my micro sim card and we are good to go. Except... I wasn't getting any SMS anymore.
After some digging, i found out that a nice feature of iOS is how a problem: if an apple device is sending an SMS to another apple device, they chose to use "iMessage" as protocol, not a regular cellular-data bound SMS. Supposedly there is a fallback if Apple can't deliver the message, but in my case I still have an IPad and OSX running, so they technically CAN deliver the message.
So, you have to configure your mobile-number to not accept iMessages anymore.
Attention: just disabling the account, disabling the phone-number or even deleting the phone number on the Apple account website is not enough: You need to fully log out on all your Apple devices from iMessage and FaceTime:
Go to Messages -> Send & Receive -> click on your Apple ID --> Sign Out
If you do this on all your devices, you finally can receive SMS again.