You can only have 1 source at a time so no, you won't get both the tv and airplay on the speakers at the same time.
Use wiim home app to switch input. (I don't think you can use the mini's buttons to cycle input.)
You can also set up auto sensing of aux input, when the mini is idle.
Not tried this myself, but I assume that wiim's use of the word 'idle' in the audio input menu refers to the Standby mode menu opttion..
i.e. if the mini goes standby/idle then a signal from the tv should auto switch the mini to the aux input.