Virginia was founded under the direction of the Virginia Company, which had two branches: one in London and one in Plymouth. The branch in London was granted control of the southern lands that England laid claim to on the North American continent, while the branch in Plymouth was granted control of the more northern lands.

After Virginia was founded at Jamestown, the company hired a crew under Captain John Smith to survey the North American coastal lands that England had claimed. This resulted in a book that Smith published in 1616 called A Description of New England which contained descriptions and many maps of both the Virginia area and the Massachusetts area. Although it was Smith who coined the term "New England," even at that early stage, official credit for the name was given to Charles, Prince of Wales, the King's son, who would be crowned king himself as Charles I the following decade.

Smith used both the terms "Virginia" and "New England" in his book, but they were somewhat synonymous. If there was a distinction to be made at that point, it was that "New England" referred to all the land in North America that England claimed, while "Virginia" referred specifically to the area being administered by the Virginia Company with its first settlement at Jamestown.

In March 1619-20, the Virginia Company of Plymouth then applied for a second charter, to establish a second colony. The petition used the name New England. Specifically, the petition sent to King James I was entitled "Petition for a Charter of New England by the Northern Company of Adventurers", and included the following text (spelling modernization and emphasis mine):

"The most humble petition of your Majesty's counsel for the second colony, and other adventurers in the Western parts of England for the plantation in the North Parts of Virginia in America...

"Whereas it pleased your Majesty...to give license for the establishing of two Colonies in Virginia in America, the one called the First Colony undertaken by certain noblemen knights and merchants about London; the other called the Second Colony likewise undertaken by certain knights and merchants of the western parts [i.e. Plymouth, England]...

"...Your Petitioners do in all humility desire that your Majesty will...First, that the territories where your petitioners make their plantation may be called (as by the Prince His Highness it has been named) New England, that the bounds thereof may be settled from 40 to 45 degrees of Northern latitude..."

King James I approved of the application, and granted the charter, in which he referred to the new colony to be settled as "New England". Famously, the first ship of settlers had already left aboard the Mayflower because they were afraid the King wasn't going to grant the charter, so they left before they could be stopped, and wrote their own charter on the way over (the "Mayflower Compact"), which became their basis of government while the King's charter was ignored when it finally made its way over to New England.

Thus, the first colony then became known as "Virginia" or "Jamestown" while the second colony became known as "New England" or "Plymouth" just as soon as the second colony was settled. As additional colonies were established near Plymouth in Rhode Island, Connecticut, and Massachusetts Bay in the following decades, these colonies collectively came to be referred to as "New England". This name was reinforced in 1686 when James II tried to re-organize all the New England colonies as well as New York and New Jersey under the administration of a new entity called the "Dominion of New England". This didn't last long--the people of Plymouth didn't recognize the King's authority to do this without their permission (the King wasn't a party to the Mayflower Compact, after all), so they kidnapped the Governor and overthrew the government. James II was deposed during the Glorious Revolution and replaced by his daughter and son-in-law William and Mary. William and Mary re-re-organized New England into the more familiar colonies that would last until the Revolutionary War.

So the reason Virginia isn't part of "New England" is because the first two successful English colonies in North America were officially founded under the authority of the Virginia Company, who named the first one "Virginia" and the second one "New England".

As for your second question about the "Commonwealth" name...After Charles I was deposed during the English Civil War, the new English government under Oliver Cromwell became known as the "Commonwealth of England". A commonwealth meant a government of the people rather than of a monarchy. So, this new government called their North American colonies "commonwealths" as well, including the Commonwealth of Virginia. The name went away once the monarchy was restored, but 100+ years later, when the Thirteen Colonies declared independence, the government of Virginia revived the old name, and once again became the "Commonwealth of Virginia" in 1776. Massachusetts and Pennsylvania did the same. After the adoption of the Articles of Confederation and then the U.S. Constitution, in which the colonies are referred to explicitly as "states", none of these states bothered to change their official name and have been "commonwealths" ever since. Kentucky followed their lead and also used the terminology, but from a legal standpoint, there's never been a distinction between a "commonwealth" or a "state" under U.S. law. They mean the same thing in a U.S. court of law.