But according to my knowledge since I am making asynchronous requests, the two requests should be made at the same time.

asynchronous doesn't mean that the requests are made at the same time. It means the code won't wait for the responses from these requests. Requests are made in the manner you add them in the code and responses will be asynchronous i.e the response will be received at some point in future time.

And there is no guarantee in what manner they will be received.

However, when you sue async/await, the code execution waits at await keyword for the response based on its success or failure it calls the next flow. It is nothing but syntactic sugar on promises. Under the hood, this is how your code will be executed:

exports.myFunction = async (req, res, next) => {
  if (some condition) {
     next()l
  }

  axios.get(`https://a-domain.com/url/path`).then(result => result).then(data => {
      const info = data;
      
      addDoc(collection(db, "name_of_collec"), info).then(response => {
        console.log("Document written with ID: ", docRef.id);
      }).catch(e => {
        console.error(e);
      })
  }).catch(e => {
    console.error(e);
  })

   next();
};

So, how it is possible that there is no error occurring?

It is not necessary that the error occurs when you request. If that is the case catch handler should be activated.

How is it that, sometimes the no document is added to Firestore? Like it is skipped somehow...

The reason might be info is an empty object and firestore automatically removes the empty document.

There is no issue, but there is not data info that is empty sometimes that skips the addition(it does request firestore, but firestore removes the empty docs)

exports.myFunction = async (req, res, next) => {
  if (some condition) {
     next()
  }

  try {

    const results = await axios.get(`https://a-domain.com/url/path`);
    const info = results.data;
    const is_Info_Data = Array.isArray(info) ? info?.length >= 1: Object.keys(info) !== 0 
    if (is_Info_Data) {
      const docRef = await addDoc(collection(db, "name_of_collec"), info);
    console.log("Document written with ID: ", docRef.id);
    } else {
      console.log(`Info is empty ${info} no addition to DB`);
    }
    
    next();
  } catch (e) {
    console.error("Error adding document: ", e);
    next(e);
  }

};