"Before/ while/ after ordering a taxi..." are these identical constructions?

"Before ordering a taxi...", and "After ordering a taxi..."; the noun-verb follows a preposition and is a gerund.

"Waiting for a taxi, patience is needed." Or "Waiting for a taxi, I dropped my wallet," are I think both participles, even though the noun being qualified is only explicit in the second.

In the case of...
"While ordering a taxi, patience is very important."
"When ordering a taxi, reliability is very important."

...are while and when currently treated as adverbs qualifying verbal adjectives (ptc) or are they categorised as prepositions (before, while, after) taking objects (ger)? Or is there another analysis?

I would say in "while ordering", while is part of the verbal phrase. "Ordering" is not serving as a normal verb, but a present participle, itself an adverb, but not one which modifies the noun (patience, reliability etc.) So "while ordering" works like an adverb, but does not modify the noun in either of those sentences, instead modifying the omitted pronoun "I".