Learn English – Tripped on and tripped over

It's basically the same as your other question and its answer.

In all your examples here, at indicates an instantaneous event (not related to anything leading up to that event) had an effect; from indicates that the effect stemmed from an existing or long-standing condition.

I became hungry at the smell of the cookies.
The smell of the cookies reached you and you became hungry (instantaneous).

I was exhausted from working all day.
Working all day is a long-standing condition, not instantaneous.

I got excited at the sight of her face.
You saw her face and in an instant became excited.

The last one is more interesting:

I'm broke from having spent all my money at the bar.

Although you could have spent all your money in one transaction, so that you instantaneously became penniless, the use of the continuous having spent means that the state of penury has continued since that time. Consequently you use from.

You could even make a more ambiguous case with

I'm broke from all the expenditure I incurred.

But the same analysis applies: you incurred all the expenditure, and an hour later you had still incurred all that expenditure. It's not the same as suddenly seeing someone or smelling cookies.

A phrase using the verb "to ban" with a subject and a location typically follows the form "To ban subject from [(verb)ing at] location"

ILLOGICAL: There’s no necessity to ban prepositions from using (them) at the end of sentences.

When you ban something from (verb)ing at location, you prohibit the subject from performing the action at the specified place. Here you are prohibiting the prepositions themselves from using (something unspecified: them) at the location the end of the sentence. The issue here is less the structure of the sentence than the choice of verb: 'using' appears to have no object, and it seems unlikely that a preposition would be able to 'use' (take illegal drugs) in a colloquial sense.

CORRECT: There’s no necessity to ban prepositions from being at the end of sentences.

Here, prepositions are prohibited from performing the action, being, when at the end of sentences. They are not allowed to exist at the end of sentences. This is quite clear.

ALSO CORRECT: There’s no necessity to ban prepositions from the end of sentences.

When banning someone/something, the "verb + at" is optional. If excluded, as in "To ban subject from location, "being in" or "being at" is implicitly understood. This example has the same meaning as the previous example and is also very clearly phrased.

Both correct examples are fairly standard and including "being at" shouldn't sound particularly strange to a native ear. Since language tends towards simplicity, however, the exclusion of the verb is more likely.