Does the silver axe break in Animal Crossing New Leaf

According to the Animal Crossing wiki:

To upgrade to T&T Mart, the player must have spent 10,000 Bells towards their House Mortgage and 12,000 Bells on items at Nookling Junction. The player must also wait for 10 days after the Town's creation before the option to upgrade is made available.

So spend lots of money in the store, and start paying off your mortgage. That will upgrade the store.

You have to get the second best store.

from Gracie's page on the wiki:

After the player has expanded the Nookling store in their town to T.I.Y., Gracie will start to visit the town's Plaza on completely random days. However, Gracie may rarely appear for some Players, making it more difficult to get the T&T Emporium.

In order to get T.I.Y.:

To upgrade to T.I.Y., the player must spend a total of 50,000 Bells in Super T&T and 21 days have passed since the opening of the store.

Also from the wiki.