Pairing Apple TV with iPad

Yes, you can connect an iPad over wireless to a wired Apple TV, if the router ‘bridges’ the two network parts as one logical network, which seems to be default for most home routers.

I don't know how to determine which Airport Express model I have. Nothing on the case and Airport Utility doesn't help - does show serial no. and firmware version (7.8).

Identifier numbers are often printed on the bottom or on the back, together with serial number, and regulatory symbols.

Firmware 7.8 is cool, as that gives AirPlay 2 capability to a discontinued device.

The important thing is that it is configured to join an existing network, or have one subnet in use for the whole network.

I was able to get the Apple TV app on the iPad, however, by installing iTunes remote app. That gave be the Apple TV app once I had used the code in iTunes on the iMac. Not sure how that happened, but the Apple TV app seems to work on the iPad after a quick check.

It may have been that the code unlocked what was previously blocked?

The apps and their names can be a bit confusing:

• The iTunes Remote app allows you to control iTunes on a Mac or PC from your iPhone or iPad, including playback to an AirPlay speaker such as whatever is connected to the Apple TV. [The older version had some capabilities to control Apple TV, but that is much more separated now.]
• The Apple TV Remote app allows you to control the Apple TV from your iPhone or iPad.
• Similarly, the Apple TV widget in Control Center on iOS 12 on your iPhone or iPad lets you control the Apple TV (4th generation or later) without installing additional software.
• The Apple TV app (has little to do with Apple TV per se) allows you to see movies, TV shows, and videos, including AirPlay to an Apple TV. The app is not available in all regions; those other regions keep the Videos app.

AirPort Express is a Wi-Fi base station that relies on a separate gateway-router for internet access. It is deferent in design from many home routers that simultaneously act as a gateway. AirPort Express can have several configurations that effect its capability (e.g. new network or joining existing network).

AirPort Express has had three different models (A108x, A1264, A1392) with varying capabilities. Which is yours?

Apple Support - Manuals - AirPort Express