Bluetooth Functionality

Would it be possible to pair the remote with wireless Bluetooth headphones to allow for listening (for example) to a TV through the paired headphones?

How is that supposed to work? The audio output of your TV does not go to the remote.

This is something that the Roku remote does. See here and here.

The headphone is paired to the Roku device. You should still be able to do that as is even when using another remote instead of the Roku remote.