They are called Participle clauses:
Participle clauses are non-finite clauses. They use a present participle or a past participle to shorten a main clause. Participle clauses are common in written English because they allow us to include information without making long or complicated sentences.
The subject of the matrix clause is the subject of participle clause too (the participle clause does not have its own subject).
We can use them to shorten active and passive sentences.
The first sentence uses the present participle (ing-form) to show that two actions are taking place at the same time:
Holding the hair-dryer in her left hand, Susan cut her hair with the scissors in her right hand.
Long form: Susan was holding the hair-dryer in her left hand and cutting her hair with the scissors in her right hand.
Comparatively, in your example, the present participle is inverted (but in both examples the present participle governs that clause):
The factory produced a foul smell in the air, angering numerous residents in the neighborhood.
We can simply switch these clauses around and they would mean the same thing:
Angering numerous residents in the neighborhood, the factory produced a foul smell in the air.
The long form for both sentences would be:
Long-form: The factory was a producing a foul smell in the air and angering numerous residents in the neighbourhood.
We use the past participle to shorten a passive clause.
Blown to the right by the hair-dryer, her hair could easily be cut.
Long form: Her hair was blown to the right by the hair-dryer and could easily be cut.
We use the perfect participle (having been + past participle) to stress that the action in the participle clause took place before the action in the main clause. However, this form is rarely used.
Having been cut, her hair looked strange.
Long form: After her hair had been cut, it looked strange.
Your second example
While in the second example both passive (1) and active (2) shortened forms are used:
(1) Raised hundreds of thousands of dollars, (2) helping numerous non-profits achieve their goals.
The first clause is a shortened passive past participle clause meaning:
(1) Long form: Hundred of thousands of dollars were raised and helped numerous non-profits achieve their goals.
While, the second clause can be analysed as a shortened active present participle clause:
(2) Long form: Helping numerous profits achieve their goals by raising hundreds and thousands of dollars.