Short answers, pulled from the story.
Banns of marriage are a public announcement, made in a Christian parish church or town council, of an impending marriage between two named persons. Their purpose is to allow anyone who knows of a legal or canonical impediment to the marriage, such as a prior undissolved marriage, a vow of celibacy, or a prohibited family relationship, to raise an objection before the wedding takes place.
The Catholic Church first required banns of marriage under Canon 51 of the Fourth Lateran Council in 1215. The Council of Trent made the requirements more precise on the 11th of November 1563, specifying that announcements must be made by parish priests of both parties on three consecutive holy days before the wedding.
After England's Clandestine Marriages Act 1753 made banns or a marriage licence legally mandatory, couples who wanted to avoid the three-Sunday public announcement had to leave England and Wales entirely. Scotland, and particularly Gretna Green as the first village over the border, was the customary destination because Scottish law did not impose the same requirements. That destination became less attractive after 1856, when Scottish law was amended to require twenty-one days of residence.
Yes. In the Canadian province of Ontario, banns proclaimed openly during divine service remain a legal alternative to obtaining a marriage licence. Manitoba also allows banns read once in a church attended by both parties to substitute for a licence. France requires civil banns to be displayed at the town hall ten days before a wedding.
When Ontario was not yet issuing marriage licences to same-sex couples, the first same-sex marriages in the province were performed using banns instead. Those marriages were ruled valid in 2003 because banns remained an independent legal basis for marriage in Ontario, separate from the licence system.
The Book of Common Prayer 1662 form reads: "I publish the banns of marriage between NN of... and NN of...," followed by an invitation to declare any cause or just impediment, and closes with "This is the first, second, or third time of asking." Common Worship 2000 offers an alternative asking whether anyone knows any reason in law why the couple may not marry. Both forms received equal statutory authority when royal assent was given to the Church of England Marriage (Amendment) Measure on the 19th of December 2012.