As of now, they can only get drunk as a result of the game 'Truth or Dare or Shot' in the Evening Living Room. You MUST play the game, though. If you click "Just enjoy the game until night", they will stop before getting drunk.
Nope, just play through the 'Truth or Dare or Shot' game and, once you have no more turns left, they will automatically be drunk. You press "Nothing more happens", and the next scene will prompt you to the "drunk options".
I remind you that new scenes may not work with old saves. This one likely doesn't.
The web (public) version, as you can see from the sidebar, is prior to the one where the scene was added. Therefore, the scene is only present in the downloaded files that are at least v 0.81.0, and are only accessible to backers.
P.S.
It mean you must unzip the zip file and double click the html file or, if you are on an Android phone, install the apk file. This way, you can play the latest version.