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Hi there. I have some question which I would like you to help me with. The emboldened part is my variant.

We reached the next town having been hopelessly lost more than once / having got hopelessly lost more than once.

Apparently, his explanation has been felt to have been less than satisfactory /  (is) was felt to have been less than satisfactory.

It wasn't until he mentioned the conference that I and he had met before / realized that we had met before.. If it were - It wasn't until he mentioned the conference where I and he had met before - would it be right?

She proceeded to fix the faulty wiring, having first the electricity been switched off / having made sure that the electricity had been switched off.

Don't worry. Everyone will have forgotten all of /all about it by the time you see them again. Is it alright?

It is six o'clock: I imagine they will have been heading/will be heading for the airport by now.
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FandorinHi there. I have some question which I would like you to help me with. The emboldened part is my variant.

We reached the next town having been hopelessly lost more than once / having got hopelessly lost more than once. okay

Apparently, his explanation has been felt to have been less than satisfactory / (is) was felt to have been less than satisfactory. okay with right context.

It wasn't until he mentioned the conference that I and he had met before / realized that we had met before.. If it were - It wasn't until he mentioned the conference where I and he had met before - would it be right? Your format is totally confusing here. The position of the main verb, "realized," cannot be determined. What is your intention? I suggest you write out each of the three versions completely.

She proceeded to fix the faulty wiring, having first the electricity been switched off / having made sure that the electricity had been switched off. the electricity first having been switched off OR the electricity having first been switched off

Don't worry. Everyone will have forgotten all of /all about it by the time you see them again. Is it alright? You can say, "will have forgotten all of it by the time etc." Your format/intention is not clear!!

It is six o'clock: I imagine they will have been heading/will be heading for the airport by now. will have headed In most cases, your use of the perfect is more complicated than the originals, and would need some justification. Your substitutions generally work, assuming the proper context.
The biggest problem for me was trying to figure out exactlywhat was being substituted for what. A couple of words more or less here and there can make a big difference!! Emotion: wink

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Hi Avangi. Sorry for being unclear.

It wasn't until he mentioned the conference that I and he had met before.
It wasn't until he mentioned the conference that I realized that we had met before.
It wasn't until he mentioned the conference where I and he had met before.

She proceeded to fix the faulty wiring, having first the electricity been switched off .
She proceeded to fix the faulty wiring having made sure that the electricity had been switched off.

Don't worry. Everyone will have forgotten all of by the time you see them again.
Don't worry. Everyone will have forgotten all about it by the time you see them again.

It is six o'clock: I imagine they will have been heading for the airport by now.
It is six o'clock: I imagine they will be heading for the airport by now.

AvangiA couple of words more or less here and there can make a big difference!!
I promise to be perspicuous next time. Emotion: smile
Fandorin It wasn't until he mentioned the conference that I and he had met before.

It wasn't until he mentioned the conference that I realized that we had met before. The second "that" is optional. Personally, I'd omit it.

It wasn't until he mentioned the conference where I and he had met before. "It" has no antecedent. We must assume that prior context provides it. Otherwise, we take "it" as a dummy, and the "that" clause becomes necessary.
(It wasn't until X that I Y'd.)
OR When did you recognize him? (reply) It wasn't until X.
As a courtesy, we usually mention ourselves last: "He and I."

She proceeded to fix the faulty wiring, having first the electricity been switched off . This is ungrammatical.
She proceeded to fix the faulty wiring, having first switched off the electricity.
OR She etc., the electricity having first been switched off. (or "first having been")

She proceeded to fix the faulty wiring having made sure that the electricity had been switched off. This is okay. I think it needs a comma after "wiring." Also, simple past works: "the electricity was switched off.

Don't worry. Everyone will have forgotten all of by the time you see them again. This works if you remove the "of," but it's uncommon.

Don't worry. Everyone will have forgotten all about it by the time you see them again. Okay.

It is six o'clock: I imagine they will have been heading for the airport by now.
It is six o'clock: I imagine they will be heading for the airport by now.
"Heading" has two senses here. One is to "change your heading," or "begin to travel toward"; and the other is "to be travelling toward." "Headed," or "to be headed," on the other hand, only has the second sense.
To me, your first version is wrong, or at best, unnatural. Others may disagree.
I think you have to choose between "will be doing by X now" and "will have done X by now." I don't think "will have been doing X by now" works. (You can say, "will have been doing X for two hours by now.")

It is six o'clock. I imagine they will be headed for the airport by now.

they will have headed for the airport by now.
they will be heading for the airport by now.

they have headed for the airport by now.
They are headed for the airport by now.
They are heading for the airport by now.
They will have been heading for the airport for ten minutes by now.