You don't have to have a wired (ethernet) connection at all for AirPlay (hence the "Air" part of AirPlay), to setup/use.
You do have to have an internet connection (however you get it) to your Roku device for the initial RokuOS firmware update and the AirPlay (hidden) app update that enable the functionality (and for any subsequent updates to either required).
As to whether internet access is required during AirPlay sessions: no, it isnt; but considering that most AirPlay sessions will probably be mirroring/casting internet content streaming to another device, it becomes mostly moot.
No idea (I don't own any Apple products), but it suggests it needs ethernet access always, and it also suggests a bug (that should be reported to Apple), because obviously many/most Roku devices are going to be connected wirelessly.
I enabled WiFi on my iMac, went to my Roku Ultra, turned on Airplay, chose my Roku from the list of available devices on my iMac Airplay menu, authorized for the first connection via a PIN and it worked as expected.
Both my iMac and Roku Ultra were connected via the same Wifi frequency (5 GHz).