Translation:Sometimes our professor speaks fast.
If that's true, then the sentence seems to have an error, and should read "A veces nuestro profesor habla rapidamente."
However, Spanishdict.com does indicate that "rápido" can be an adverb or adjective. Hence I'm concluding that the lesson here on adjectives, perhaps as Snoue is suggesting, is to clarify that rápido can be an adjective, yes, but it can also function as an adverb.
no, in this case rápido is an adverb, modifying the verb hablar, so the gender or the number of people speaking doesn't have an influence on it: anyone speaks "rápido". if you were to use it as an adjective that described the speaker instead of the way they talked, then yes, you would confirm it to gender and number and use "rápida" for "profesora".
Rápido y sin distinguir las vocales. I listened to that once and wrote 'nuestro profesor', then listened again to check and decided it was 'nuestra profesora' and no, it was 'nuestro profesor'. The 'habla' eats the obvious distinction. Exactly the same thing in the last sentence with 'secretario/ secretaria' too. Bear in mind that in normal listening contexts people KNOW if the person's a man or a woman, they don't have to rely on hearing the distinction. You could be a bit kinder - or clearer - in these situations.