We don't have ballot initiatives at the federal level. The US is a democratic republic, so it's a bit pedantic, but if you had to classify as one or the other the U.S. is a republic.
No, that's not what a republic means. The U.S. is a republic. It is mentioned explicitly in the U.S. Constitution in Article 4, Section 4:
"The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened) against domestic Violence."
As mentioned, the definitions of republics and democracies have blurred over the years as people often use them interchangeably. Republics have representatives. Just as the ancient Roman Republic did, with its famous Senate. Pure democracies run on majority rule and direct voting on laws.