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.
I don't believe that "for many times" is grammatical in any variety of English.
(Well, unless you contrive an example where "times" is the plural of "a given time of occurrence" and there's an elided "of the": "Class times are listed below. For many times, you can click the link for more information.")
You can use "for" with an ordinal: for the first time, for the 20th time, for the last time. Notice that it's always time, not times, in these examples. You can also use "for" with a duration: for 100 years, for the summer, for a long time.
"Many times" behaves the same way as "100 times" or "just a few times", and adding a "for" is not grammatical.
Mark has been to Hong Kong many times.
Mark has been in Hong Kong for two years now.
After Jessica failed 10000 times, she finally created a working robot.
After Jessica failed for the 10000th time, she tried a different approach.
Best Answer
She was one of many can stand on its own, meaning that there were many other people in a similar situation.
She was one of the many normally requires a continuation to state what that situation was, for example "She was one of the many who were finding life difficult at that time".