That is correct, although I would consider "fateful" more idiomatic.It seems the second meaning of 'fatal' and the word 'fateful' are interchangeable. In other words I think the word 'fateful' in 'the fateful day of President Kennedy's assassination' can be replaced by 'fatal'. Is that right?
As for your other question,
...very serious and having an important bad effect in the future
The underlined part simply describes when will the bad effect happen
having an important and usually negative effect on the future
Here, the underlined part describes on what will "the thing that is fateful" have a negative effect. You know, it's like in other phrases with "effect on", e.g. This noise will have a negative effect on your hearing
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