It depends on which congress you are referring to and which government.
Regulation of commerce
No. Congress does not have the power to establish a minimum wage at all.
Yes Congress can regulate the minimum wage. Congress also created the federal minimum wage in 1938. The current minimum wage is $7.25.
government power in an economy.
No.
The power to pass a national minimum wage law
yes they can its an implied power
Congress can declare war (senate) and ratify treaties.
The power to pass a national minimum wage law.
The power to pass a national minimum wage law
Congress has the power to tax
constituent power of congress