There didn't exist computers.
Will there happen similar events in the future?
Are these sentences grammatical or is my mind making stuff up? I haven't been able to find much on the net.
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I could be wrong, but I do not think there is anything grammatically wrong with your sentences, however, it is extremely unlikely that a native English speaker would ever phrase a sentence in this way.
For the first sentence, I would expect to hear something like,
'Computers did not exist'.
Even this phrase is very terse and would probably be expanded to something like,
'Computers did not exist at that time',
'This was a time before computers were invented'.
For the second sentence, I would expect to hear something like,
'Will similar events happen in the future?'
'Is it possible that similar events will happen in the future?'
There can be used as a pronoun subject of a sentence (and so before a verb)
Most commonly it is used in the form "there is" or "there are".
However it can also be used with other verbs expressing existence:
In the forest there lived a woodcutter.
There remains the question of the budget.
On the mat there sat a cat.
These are slightly "elevated" in style. For example the first sentence could be expressed "A woodcutter lived in the forest". Using "there" makes this sound like the start of a fairy story.
It is not generally used with other verbs. For example. "To the bus stop there ran James" would be odd. In the correct examples "lived", "remains" and "sat" all include the meaning of "exist in certain place or time", but the verb "run" suggests an action, not existence.
So while "There did not exist computers" may be correct, it is not the common way of expressing this idea.