I would like to know if, in addition to any MEL requirements, there is any regulation or requirement about having an operational TCAS II while operating in US airspace?
TCAS II is required for most airline flights. This is not a requirement for certification, and the TCAS may or may not be on the MEL. However, most flights under part 121 (scheduled airline service, as well as some freight) require TCAS II.
As noted in FAR 121.356, "Turbine-powered airplane of more than 33,000 pounds maximum certificated takeoff weight" must be operated with equipment that meets a list of standards, but is essentially equivalent to just requiring TCAS II. Smaller aircraft with 10-30 seats can use TCAS-I.
No. It's not required in the United States. There is nothing in Part 25 of the regulations. However, if you install one it should comply with this AC.