"Why do Cardiff, Swansea, Wrexham, Newport, Colwyn Bay and Merthyr play in the English football league?*"
In the case of the classic big three of Cardiff, Swansea and Wrexham, these clubs are amongst the earliest in the entire Kingdom, let alone in Wales. Wrexham date back to 1864, making them Wales' oldest and supplanted only by Sheffield, Cray, Notts County and Bradford PA, and are further the third oldest professional club in the world.
Given this, when leagues first came about, it made much more sense for these teams to play across the border, because there was greater competition than playing friendlies against village teams. I don't know about the other clubs exactly, but Wrexham played in the Birmingham/Midland combination league for about 30 years until joining the Third Division in 1921 and stayed in the football league for 87 years.
Bangor City used to play in England, too. They were founding members of the Conference, or Alliance League as it was then known. It works the other way though; Northumberland's Berwick play in Scotland, Northern Ireland's Derry play in the Republic and Shropshire's Oswestry is the home of the 'Welsh' team The New Saints. And right up until 1990, teams like Kidderminster, Hereford, Shrewsbury and Chester competed in the Welsh Cup.
*Given that the Premiership is technically not a member of The Football League, 2013/14 will be the first season in roughly a hundred years to feature no Welsh teams in The Football League, and not the football league system. Big difference.
As for the League of Wales, the most famous product to come out of it is probably Owain Tudur-Jones. It's a terribly mismanaged league, thanks to the geniuses in charge of the FAW. Only 12 clubs after downsizing from 18. But still, come on you Nomads!