I think ''Лежать'' is commonly used when sitting or being on a specific surface, for example: ''Я хочу лежать на кровати,'' meanwhile "Стоить," is often used when being standing at a certain place, for example: ''Я не люблю стоить у бабушки.'' »» Please, someone correct me, if I'm wrong!
The original comment by Karadaieva was a joke. Many students are supposed to party a lot and drink a lot, so the joke makes a reference to this fact by suggesting that a student is more likely to lie under a bridge (because they're so drunk and fell asleep on their way home) than to stand there.
Unfortunately, your description is not correct. The difference between лежа́ть and стоя́ть is pretty complicated. In general, стоя́ть refers to 'being placed vertically', while лежа́ть is 'to be placed horizontally'.
For example, кни́ги стоя́т describes a situation when books are placed vertically on a shelf:
Кни́ги лежа́т describes a situation when book are piled one above another, probably on the table:
However, the objects that are normally placed horizontally might be used with стоя́ть even when they're placed horizontally. So, стоя́ть might refer to the default, 'correct' position of the object, while лежа́ть would mean it fell somehow.
We don't use стоя́ть for 'stay'. We'd say «Я не люблю́ остава́ться у ба́бушки». Also, mind that стоя́ть 'to stand, to be (in some place, positioned vertically)' and сто́ить 'to cost' are different verbs.
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'.