Why is the rotation gizmo not working in 2.74? When I left-click using the Translate or Scale gizmo it translates or scales according to axis, as expected; however, when I left-click close to the colored circles of the rotation gizmo, it doesn't rotate the object and just moves the 3D cursor. 

Is it a bug? If yes, is there a standard way of reported such bugs (I've seen the blender.org/forum is closed)
It's not a bug, you can imagine this widget is something that's used by an incredible amount of people and is heavily tested. But this issue has cropped up from time to time and depends heavily on the graphics card drivers used. 

Try Setting "Selection" in User Preferences System Tab from "Auto" to "OpenGL Select" or "OpenGL Occlusion Queries"

The keyboard shortcuts for the transform tools are G, (grab), R (rotate), and S (scale), often they are more convenient, in combination with their follow up keys x, y, z to select the axis..