Yes, the Earth transfers heat from the surface to the atmosphere via conduction and some radiation.
Yes, the tropopause stops convection due to temp inversion.
Yes, as temp (delta T) increases, heat transfer increases (all else being equal). Yes, that heat takes all paths in proportion to their resistance.
The average surface temp is 15C and tropopause is -56C. (taking your word) This is probably because air pressure drops. The universal gas law (PV = nRT) shows that as pressure drops, temp drops in proportion. With a smaller column of air above it, the higher altitude air is under less pressure from above (lower P) so it expands some (smaller increase in V) and so T must drop.
Which is more efficient between convection and radiation? That depends on the situation. Radiation goes right through a vacuum while it stops convection. Low temps have a dramatically lower heat transfer rate than higher temps so convection may win out if there's a free gas involved. Either way, radiation has by far the more magnitude of change when surrounding conditions are changed by a small amount.
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