What affects the different types of luck in Animal Crossing New Leaf

According to the wiki

During the summer, (the second half of July–September) it is possible to get a tan, as real humans would. In order to do this, the player must stand in the sun long enough (note that there must be a clear sky outside and it must be during daytime), the player's skin will go up through levels. There are five different levels, each one having a darker hue than the other. Unlike the older games, City Folk and Wild World, you start to notice the tan right away, but you can only achieve 1 level per day.

The player can also get a tan anytime in the year, as long as they visit Tortimer Island in the morning. The only thing that stops a tan from appearing is an umbrella.

Yes. The silver axe can eventually break.