What you must keep in mind is that "while" doesn't only carry the literal meaning of "during a period of time".
For example, the first sentence would read completely differently depending on whether you use "while" or not.
"Being a good person, he devoured kittens" means that he devours kittens because he's a good person. Compare, for example, "Being stubborn as always, he disagreed with us" - it means that he disagreed with us because he was stubborn.
"While being a good person, he devoured kittens" means that the person is generally good, but despite that they devour kittens. Compare "While being stubborn as always, he could see our point" - he was still stubborn, but despite that he agreed to something.
In the second sentence, "while/when" might technically be correct, but it reads awkwardly - something like "you agree to our terms as long as you're using the services". I'd rather use "by using" here - meaning that "starting to use the services carries an implicit agreement to our terms".
In the third one, "during providing" is ugly - you generally avoid two -ings in a row. Otherwise, I think both are fine, though "while providing" reads better to me personally.
The fourth is a bit similar to the first - the meaning flips when you apply "while" to the sentence. So either this low must be applied because it's the main instrument etc. (without "while"), or despite that (with "while").
According to the British Council the pattern having been + past participle is not the past perfect passive but a perfect participle which indeed indicates the order of events:
If we wish to emphasise that one action was before another then we can use a perfect participle (having + past participle):
Having won the match, Susan jumped for joy.
Having been told the bad news, Susan sat down and cried.
This said, the test is correct, that is, the perfect participle is improperly used. In terms of grammar this source says that it's called perfect participle, and of course, in your case, passive.
Best Answer
It's a bit controversal question. Some grammarians say that the present perfect continuous tense doesn't exist in the passive form. They say that the nearest equivalent of a sentence such as They have been repairing the road would normally be The road has been repaired lately. They have been constructing the building for four months would be The building has been under construction for four months.
Others say that sentences in all tenses can be formed in the passive forms; the present perfect continuous is no exception. However, its use that is less common should be avoided. According to them, it should be used to express events recently occurred in the past and continue in the present. For example, The toys have been being broken by children. Too much pollution has been being dumped in the river.
In light of the above, I would go for the sentences presented by TRomano in his comments. Alternatively, you can start your sentence with "being" or "having been" as suggested in the question.