Both phrases imply that they were at some point going to exist (the "yet" makes that explicit, even if it weren't implied otherwise). The difference actually has to do with what happened after they did come to exist.
The term "had existed" actually implies that something at one time existed but does not anymore. Likewise, "had not existed" implies that they had not yet come to exist, but then later they did exist, but then after that they ceased to exist again.
On the other hand, "did not exist" is more open-ended, it only says that they didn't exist at that particular time (but usually implies by context that they did exist at some other time). "Did not exist yet" clarifies this to mean that they did not exist at that time, but would come to exist later (but it does not put any end to that condition, so they could continue to exist now).
If "national skinheads" no longer exist now, then it would perhaps be technically correct to say "they had not existed yet", but this is a rare enough (and very specific) situation that the construct is almost never used, and therefore even in this case it would probably sound awkward to most listeners.
Let's first talk about the following two sentences:
1- Sara went to bed as soon as she had finished homework.
2- Sara went to bed as soon as she finished homework.
I think your confusion is valid because we use the past perfect when we talk about something that took place before another thing in the past. So the use of the past perfect comes across in the first sentence but the use of the past simple in the second sentence doesn't. Am I right? In fact, we don't need to use the past perfect unless it is necessary or unavoidable to do so. Even if we talk about one action happening before the other one, it is possible to use the simple past for both actions if we think it is not necessary to highlight or emphasize the happening of the earlier action. It sounds natural to avoid using the past perfect where the simple past works, which is used to refer to something or several things happening in sequence (one after another) in the past.
So both of the sentences are grammatically correct. However, I'll prefer the second phrase to the first one.
As for the last two sentences, it is correct to say that "everyone had gone home when Sara got to the party", but it's not grammatically correct to say that "everyone had gone home when Sara had got to the party". It doesn't make sense. In the past perfect when we talk about two events, we use the simple past in one clause and the past perfect in the second clause.
Let's now talk about the following sentence you are confused about:
"Everyone went home when Sara had got to the party".
There is nothing wrong with this sentence, but the meaning is other way round. It means that first Sara got to the party and then every one went home. Look at the
first sentence again. When Sara got to the party, everyone had gone home. Here it means that first everyone went home and then Sara got to the party. Sometimes, one action happens soon after the other action, here we should use the past simple in both clauses such as when Sara got to the party, everyone left, when they saw the police, they ran away, etc.
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Best Answer
Once they passed the exam, they worked in a national hospital and received more training
The sentence is grammatical. Another example:
Once he got inside the house, he slammed the door (The Free Dictionary).
The word once can be used as an adverb and a conjunction. As an adverb, it implies one single time. When you use it as an conjunction, it means as soon as or after. You can use the simple past or the past perfect in the "as soon as/after" clause. Same is the case with the conjunction "once".