What are some "American rules" that Waverly's family adopts?

The Waverley family also adopts other American rules such as giving the children American names.
For example, Waverley's mother names her daughter Waverley Place Jong, after the street the family lives on. Waverley uses this important name on official American documents. Her mother likely chose the name because it follows the American tradition of having first and middle names.
The family also adopts the way Americans calculate age. In the text, Waverley tells us that, when the Santa at the First Chinese Baptist Church asks her how old she is, she answers that she is "seven according to the American formula and eight by the Chinese calendar."
Waverley's family also adopts another important American "rule": the importance of discovering truth for oneself. This spirit of independence is greatly cherished by Waverley's mother. When Waverley complains that the rules of chess are confusing, her mother tells her to make sense of it for herself. So, Waverley makes her way to the Chinatown library and borrows some books about chess. Then, she sets out to learn the rules, strategies, and tactics of the game. By doing so, she becomes a national chess champion at nine years old.

Arguably the biggest "rule" that Waverly's family adopts within the story is that of the pursuit of the American Dream. This rule manifests as the firm belief that if an individual only works hard enough and persistently enough, she can achieve whatever she wants or be whoever she wants. It is a philosophy firmly rooted in the optimistic nature of Americans. We see this "rule" manifest when Waverly's mother insists that Waverly can be the best at anything and that she may even become a prodigy. 
On a more literal level, Waverly's family gives their children American names and makes large efforts to fit in with American customs, including attending church and celebrating Christmas. They also make room for a shift in gender dynamics, with Waverly's chores (a traditionally feminine duty) being passed along to her brothers in order to make more time for her chess practice. 

Waverly and her older brothers adopt American rules when they visit the First Chinese Baptist Church to receive Christmas gifts. When Santa asks Waverly if she believes in Jesus Christ, she says yes so she can receive her choice of the Christmas gift boxes the chuch is giving out. When Waverly's brother, Vincent, receives a used chess set with missing pieces, Waverly's mother tells him to throw it away. Instead, he disobeys, and the children keep the chess set. 
When Waverly begins to play in chess tournaments against strangers, she is following American rules, not Chinese rules. As Waverly says, "They would have American rules. If I lost, I would bring shame on my family." The idea of playing against strangers and possibly bringing shame to the family is American, but Waverly's mother relents and lets her daughter play in the tournaments. After Waverly becomes a chess prodigy, she violates many Chinese rules. She is allowed to practice chess all the time and not do work, which is contrary to the Chinese tradition of the daughters doing the work around the house. In the end, Waverly's chess playing starts her down a path of becoming far more independent than the traditional Chinese daughter and playing by American rules than traditional Chinese rules.