This question about what documentation is required can only be answered by the plan because it's up to the plan to accept the reason that you qualify to obtain a hardship distribution.
A distribution received before the date of the birth will not qualify as a Qualified Birth or Adoption Distribution. Distributions from a Roth 401(k) are a proportionate mix of contributions and earnings. The proportion that is earnings will be subject to income tax and early-distribution penalty (unless you have some other penalty exception that applies, which seems unlikely).
