No, "il est un élève" is not correct : "C'est un élève" or "Il est élève". the same for sudent : "Il est édutiant" or "C'est un étudiant". same for profession nam : "Il est boulanger(baker)" or "c'est un boulanger"
It's simply because in French one does not use an article when stating one's occupation, and, although not a job, being a student roughly falls into that category of occupations. Why? Because language is weird.
Strangely, the playback button doesn't seem to be playing the liaison anymore (in this discussion thread anyway). I imagine the liaison would be optional, but doing it does make the sentence sound similar to the past imperfect (as above).
I think this is a situation where the liaison is optional. There are three types of liaisons and they are required, optional and forbidden. If you google French liaisons you will find a lot of information about liaisons.
This would be because the "nt at the end" is not pronounced. It affects the sound of the vowel, making it more nasal than it would be if the word just ended in "a", but you only hear the "nt" if it is followed by an "e" (i.e., "étudiante" - a female student.