You have to return that Promise as mentioned here.

To return data after an asynchronous operation, return a promise.

exports.createGame = functions.https.onCall((data, context) => {
    const gameChosen = data.gameChosen; 
    const adminName = data.adminName;
    const numberOfRounds = data.numberOfRounds;
    let gameID = Math.floor(100000 + Math.random() * 900000).toString();
    var numberOfQuestions = 0;
    var questionsPicked = [];

    getGameQuestions(gameChosen, numberOfRounds).then((questionsPickedArray) => {
        //Just add a return here
        return db.ref(`live-games/${gameChosen}/${gameID}`).set( 
         {
           playerAdmin: adminName,
           game: gameChosen,
           players: [adminName],
           questions: questionsPickedArray,
           datetimeCreated: Date.now(),
         })
         .catch((error) => {
            if (error) {
              console.log("Data could not be saved." + error);
            } else {
              console.log("Data saved successfully.");
              return {
                gameChosen: gameChosen,
                gameAdmin: adminName,
                questions: questionsPicked,
                gameID: gameID,
              };
            }
          }) 
        );
      });
    });

The following modifications should do the trick:

exports.createGame = functions.https.onCall((data, context) => {
    const gameChosen = data.gameChosen;
    const adminName = data.adminName;
    const numberOfRounds = data.numberOfRounds;
    let gameID = Math.floor(100000 + Math.random() * 900000).toString();
    var numberOfQuestions = 0;
    var questionsPicked = [];
 
    return getGameQuestions(gameChosen, numberOfRounds)  // We return the Promises chain, see below for more details
        .then((questionsPickedArray) => {
            return db.ref(`live-games/${gameChosen}/${gameID}`).set(
                {
                    playerAdmin: adminName,
                    game: gameChosen,
                    players: [adminName],
                    questions: questionsPickedArray,
                    datetimeCreated: Date.now(),
                })
        })
        .then(() => {
            return {
                gameChosen: gameChosen,
                gameAdmin: adminName,
                questions: questionsPicked,
                gameID: gameID,
            };
        })
        .catch((error) => {
            // See https://firebase.google.com/docs/functions/callable#handle_errors
            throw new functions.https.HttpsError(...);
        });
});

function getGameQuestions(gameChosen, numberOfRounds) {
    var questionsPicked = [];
    var numberOfQuestions = 0;
        return db.ref(`games/${gameChosen}`).once("value")  // Here too, we return the Promises chain
        .then(snapshot => {
            val = snapshot.val();
            numberOfQuestions = val.questionsAvailable;
            for (var i = 0; i < numberOfRounds; i++) {
                let questionNumber = Math.floor(Math.random() * (numberOfQuestions - 0 + 1) + 0);
                if (!questionsPicked.includes(questionNumber)) {
                    questionsPicked.push(questionNumber);
                }
            }
            return questionsPicked;
        });
}

So, you need to chain the Promises and return the result to the client as shown above: as explained in the doc "the data returned by the promise is sent back to the client" (which is the case if we return the Promises chain) and the data shall be an object/value that can be JSON encoded (which is the case with the { gameChosen: gameChosen, gameAdmin: adminName, ...} object).

For the getGameQuestions() function, since the once() method returns a Promise, you don't need to encapsulate it into a Promise. Again, just return the promises chain composed by once().then(...).