1 yes
It depends on the software that uses audio.
A typical DAW like Cubase or Reaper has access to all possible outputs from the interface wich are in this case Software Playback channels in Totalmix.
Lots of software don't have the ability to select different outputs, they just use output channel 1/2.
2 yes, until you completed the configuration with the ASSIGN, you have to assign the headphone output to make it appear.

No need to see the PH3/4 output channel, it is listed in the ASSIGN, click the ASSIGN button, click Phones 1 and select the PH3/4

After the assign configuration this is visible :

For making the mute button on the channels working it has to be set active by clicking the M button in the control room

Hardware Output channel AS1/2 is the XLR outputs sockets at the backside of the Babyface.
For the headphones you have to use the fader of the Phones 1 channel or the Babyface rotary after selecting the headphone Out.

All Harware output channels can have the Ableton or Logic signal as you route that signal from the Software Playback channels to the outputs.
The Main-Out can be any output channel available in the ASSIGN.
Usually the Main-out got assigned to AS1/2 for connecting the monitor speakers to the XLR outputs.
That is not fix. You have to prepare your interface for your demands. What you see in the video is how they prepared it.

Click one time onto an output channel next to the fader in the field. The whole channel gets highlighted.
Now see the fader position of the Software Playback channels, fader is up means signal is send in this amount to the highlighted output channel.
Every output channel has its own configuration of fader positions (submix)
Totalmix is a monitoring programm to send music to the outputs by you.