What does Cole begin building in this chapter? Why must it be of such high quality, and why do Edwin and Garvey not help him with the labour?

Cole is building his cabin. He had burnt the last one down on purpose. Part of his "second chance" was to build the cabin better than he had before. Cole must learn to atone for his mistakes and to improve the second time around. The two elders can't help because this is part of Cole's healing process; he must take responsibility for his actions.