Can an iMac connected via Ethernet to an AirPort Extreme participate in the Wireless network?
Actually you may have a minor, if understandable, misunderstanding of how a network works.

If you plug a Mac (or a PC, or anything else...) into a base station that other computers connect to wirelessly they are all on the same network.

Neither MacOS nor Windows, Linux or anything else differentiates a wireless network with a wired network coming from the same source.

Try it out, turn WiFi off on on your Mac and then try those things out, you will find that they still work just fine.

Note that the "green bubble messages" distinguish a text message (or SMS if you prefer) that goes over only your carrier's cellular network from the "blue bubble messages" (iMessage) which use your phone's 3G or 4G networking to send data through Apple's messaging servers. This is why your blue bubble messages never get to anyone who don't have an iPhone or a Mac.