All the times when I've seen the sentence "whatever стоит wherever" it was translated as "there is a whatever wherever". Now why doesn't "there is a student under the bridge" work? Why is it a particular student?
Because word order.
«Студе́нт стои́т под мосто́м» is 'The student is standing under the bridge.' «Под мосто́м стои́т студе́нт» is 'A student is standing under the bridge' or 'There is a student under the bridge.'
See my comment here for a different example of how word order influences the meaning: https://www.duolingo.com/comment/12192662
In sentences with neutral intonation, you place the known piece information first, and then the new information. So, in «Студе́нт стои́т под мосто́м» you tell something about the «студе́нт». «Студе́нт» is someone you've mentioned before, or your listener knows which student you mean, so that's why we translate it with 'the student'.