He erases whatever he wills, and fixes.
This is grammatically incorrect, because "to fix" is a transitive verb. It requires an object: something which gets fixed. "He [...] fixes." is an incomplete sentence, because it is not apparent what he is fixing.
He erases and fixes whatever he wills.
This is grammatically correct: he fixes "whatever he wills". He also erases "whatever he wills".
With him is the original record.
This is correct but archaic. Modern English would reverse this and put the adverbial phrase "with him" at the end: "The original record is with him." The original construction is, strictly speaking, correct, but it is a more old-fashioned - even archaic - way of writing the sentence.
So, the first option is grammatically incorrect because "he fixes" is lacking an object (the thing being fixed).