Rubbing of sticks is not a pure thermal phenomenon, so it's not a question of simply two lower temperature objects achieving a higher temperature on their own. The energy input comes from the rubbing together and overcoming the induced friction caused by the pushing together of the sticks.
As for the "momentum" question, that's a complete misunderstanding of what the initial conditions are, and where the heat is stored within the system. Consider the simple case of a solid object with a constant T1 on one side, and natural convection on the other, resulting in a surface temperature of T2. There is a thermal gradient, which causes heat to flow from the T1 surface to the T2 surface. Remove the heat source on the T1 side and insulate both sides. The T2 side will increase in temperature, while the T1 side will decrease. over time, the temperature will equilibrate to (T1+T2)/2, the average temperature. Pure conservation of energy, and pure Fick's law (gradient drives flow).
In many supposed thermal momentum allegations, it's simply because people cannot observe all the state variables, and see only T2 increasing, and assume there's a "momentum" involved.
TTFN
FAQ731-376