No, this 出し is not a noun. You can say this simply because a noun does not take an object marked with を. This 出し is the continuative form (連用形) of 出す. 出す and 始める are simply connected. This has been asked many times. See:
- て versus combining-form for joining clauses
- Why do they use the 連用形 instead of the て form here?
- なく vs. なくて and stem form vs. てform as conjunctions
As for the difference between 声を出す and 声に出す, 声を出す cannot take another object because 声 itself is the object. 声に出す can take another object, e.g., 感謝の言葉を声に出す. See also: What does "声が出る" mean?