How much heating does Earth inner core provide to the surface? Compared to the energy that the Earth's surface receives from the Sun, how much power comes from the inner melted core?
How important is this contribution to the surface temperature?
 A: 
Compared to the energy that the Earth's surface receives from the Sun, how much power comes from the inner melted core?

Very little. The Earth's surface emits about 503 watts per square meter (398.2 W/m2 as infrared radiation, 86.4 W/m2 as latent heat, and 18.4 W/m2 via conduction/convection), or about 260,000 terawatts over all of the Earth's surface (Trenberth 2009). The ultimate source of almost all of this energy is the Sun.
Estimates vary on how much heat crosses the core/mantle boundary, from 4 TW to 17 TW. Even the larger value is much, much smaller than the heat emitted by the Earth's surface. Estimates of the total heat flow from the interior of the Earth (core, mantle, crust) are much tighter, 46 TW ± 3 TW (Jaupart 2007) (cf 47 TW ± 2 TW (Davis 2010)). This is considerably more than the heat coming from the core, but it's still small compared to the Earth's total heat budget: 
$$\frac{\text{heat from interior of Earth}}{\text{total}}\ = \frac{46\ \text{TW}}{260,000\ \text{TW}}\ =\ 0.02\% $$

Davies, J. H., and D.R. Davies (2010), "Earth's surface heat flux," Solid Earth 1.1:5-24.
Jaupart, C., and J. C. Mareschal. "Heat flow and thermal structure of the lithosphere." Treatise on Geophysics 6 (2007): 217-252.
Trenberth, Kevin E., John T. Fasullo, and Jeffrey Kiehl  (2009), "Earth's global energy budget," Bulletin of the American Meteorological Society 90.3:311-323.
A: The difference between the insolation power of the Sun producing minus 18.165C
(K E Trenberth et al) and Earth's lower atmospheric T measured ~15C is 33C. Therefore 33C is the temperature that is contributed by Earth's core. There is no other credible answer.
