How did westward expansion shape British North American colonies and the U.S. between 1607-1877?

That is a massive question indeed.  Another thing to consider is that fact that when Britain began expanding across the Appalachian Mountain range, it ran into trouble with the Iroquois Confederacy, and more native resistance in general in hostile terrain.  The French exploited this by arming the native tribes, and Britain attempted to restrict settlement in order to appease them after the French and Indian War.  This caused tension with the poorer Scots-Irish settlers forced to seek land in the rugged hill country.

Later in US history, also consider the connection between westward expansion, the rapid growth of slavery after 1800, and the growth in tension between southern and northern interests that ultimately led to the Civil War.  The war may well have been delayed if there had been a slower move west, or no expansion in that direction at all.