The biggest effect it had on the colonies was probably that it drove at least 3 separate waves of migration from England to America. Before the war, the bulk of settlers were puritans from east of England came and settled in the northeast. After the war, most were former cavaliers, mostly from the south of England, who settled the Chesapeake and southern coasts.. After the restoration, there was as second wave of "puritans" but this time they were mostly from the north midlands, not east Anglia, who settled Pennsylvania and the surrounding area.

These three areas all had profoundly different cultural attitudes, and settling in America made them starker, not milder. these three cultures (plus others) continue to exist in the US and have a profound influence on american culture as a whole.

Edit: For the best history of this movement, see Albion's Seed by David Hackett Fischer. If you're going to read one book about american politics, it's this one.

One way in which the English Civil War impacted the 13 colonies (really, only 7 of the 13 existed during this time) was that it aided in the creation of the New England Confederation in 1643. There was a concern in the colonies that the Dutch (in New York) or the French (in Canada) would attack the vulnerable colonies because the mother country was thrown into a state of chaos. Therefore, the colonies of Massachusetts Bay, Plymouth (still a separate colony at this time), Connecticut, and New Haven (also a second colony which is affected later by the English Civil War*). The only exception was Rhode Island, a colony the others did not like due to its religious freedom.

The Confederation, was mostly a mutual defense pact, but it also exercised other inter-colonial powers like extradition of criminals and servants and established precedents as a political organization for settling disputes between colonies. Historians consider this one of the first of many important steps in establishing a political relationship among the colonies that would eventually result in the creation of the United States.

*One of the reasons New Haven never ends up being its own colony is because King Charles II refused to grant them a royal charter partly due to the fact that New Haven provided shelter to several "Regicide Judges" that sentenced his father to death during the English Civil War

I don't think anyone answered OP's 2nd question on Cromwell's attitudes.