Using an expired SIM card in Airplane Mode
The reason it says "Sprint Wi-Fi" at the top of the display is that you currently have "Wi-Fi Calling" enabled. This means that your calls and text messages are sent via Wi-Fi over the public internet to Sprint instead of being transferred via a cell signal. This is a helpful feature when you haven't got a cell signal, but have an internet connection over Wi-Fi.


The word "Sprint" comes from the packet gateway that the phone connects to over Wi-Fi. It is not read off the SIM-card as such. Your provider Ting has a cooperation with Sprint to use the packet gateways of Sprint to enable this feature.
If you cancel your subscription with your provider, Wi-Fi Calling will also stop working. It doesn't matter if you're in AirPlane mode or not - when you try to place a call using Wi-Fi Calling it will be rejected by Ting/Sprint if you have cancelled your subscription. 
These answers are not specific to Ting/Sprint, but are true for any cell provider that supports Wi-Fi Calling.