Is there a category for Set-Asides for Emerging Small Businesses? If yes, where are the rules and regulations governing such Set-Asides?
No, there is not a category for set-asides for emerging small businesses. See FAR 19.501(c),
The contracting officer shall review acquisitions to determine if they can be set aside for small business, giving consideration to the recommendations of agency personnel having cognizance of the agency's small business programs. The contracting officer shall perform market research and document why a small business set-aside is inappropriate when an acquisition is not set aside for small business, unless an award is anticipated to a small business under the 8(a), HUBZone, SDVOSB, or WOSB programs."
"Emerging Small Businesses" was term found in the Small Business Competitive Demonstration Program, which was repealed in 2010, see https://www.sba.gov/content/small-business-competitiveness-demonstration-program