- Are each grammatically correct?
- Nuances aside, are there any fundamental differences in meanings?
Sentences #1, #2, and #4 are grammatical and mean the same thing (unless the subject implied in #4 by the context is different from 私, of course). Sentence #3 is not grammatical.
Note that 私は緊張のあまり is not a single meaningful component of a sentence in #1. In #1, #2, and #4, 緊張のあまり is an adverbial phrase and modifies 頭の中が真っ白になってしまった. The position of an adverbial phrase is often flexible in Japanese, and you can put it before or after 私は.
If anything, putting it before 私は as in #2 may put some emphasis on 緊張のあまり.
- Because the first-person is implied, there is no need to mention 私, so #4 is the most natural?
I do not think that there is a difference in naturalness between #1, #2, and #4 in isolation.
If the 私は part is obvious from the context (e.g. because the preceding sentence states 私は explicitly), stating it again is unnatural unless it is repeated for some reason.