>>347 You are using it a little wrong, let me try to explain, this is really difficult: The "so" in your sentence does not modify "that" at all, it is being used to make "hard" stronger. →"I studied so hard." -So still has the same meaning here.
But without the word "hard," you could not use the word "that" here. You would have to say: →"I studied, so I passed the exam." It is like the word "that" is used as a replacement for "so" to prevent it from being used too much. So you would say: →I studied so hard that →I studied so quickly that →I studied very hard so →I studied for a long time so →I studied all day so Only in the sentence where you already use "so" will "so" turn into "that".
You cannot change the word order like you did: →So hard did I study that I passed the exam. "so hard" is an adverb for "study" so it always has to come after "study"
For the last sentence: If you change the order of sentences of "so" it turns into "because" →I ate too much so I got sick. turns into →I got sick because I ate too much. So you would want to use: →I passed the exam, because I studied so hard"
>>349 →"She RAN so fast that nobody was able to catch up with her. "
→"So fast did she run that nobody was able to catch up with her. " This sounds like it could be from a poem, but not as proper sentence.
→"Nobody was able to catch up with her, so fast did she run." The same here.
Are you trying to learn and understand old English writing or poetry? That is what these sentences sound like to me, but people do not talk like this. It might be good to know how to read such sentences but you may confuse people if you order your sentences in this way.
Unless you got these sentences from a poem or a old piece of writing, these are not correct, and they would sound very out of place in modern English.