Quick answer: Select a search box on the TV, tap the keyboard key in Couchius, and type. Letters appear on the TV as you type, and Return submits. The apps button lists everything installed on the TV, with one tap to open.
Keyboard
On Roku and LG, each letter is sent as you type it, and deleting on the phone deletes on the TV. On Sony, the whole field is updated as you type. Either way, you skip the on-screen keyboard and the arrow keys.
Apps
The apps list comes straight from the TV, so it shows exactly what you have installed: the streaming apps, live TV and settings. Kid Mode uses the same list so a grown-up can pick which apps the four show buttons open.