Your whole argument about Wales not being a country unlike England/Scotland/Ireland is irrelevant to the debate. I'd argue against it, but this isn't the thread for it.
However, in terms of football Wales is certainly a seperate nation to the other UK nations. Our national team is fully recognised by all the relevant powers and is as valid as any other international side. Our FA runs a fully recognised national league and cups and these are again acknowledged by the various powers. So in footballing terms, we are a seperate nation, that is indisputable.
The reasons why Swansea, Cardiff, Wrexham, Newport County, Merthyr Tydfil and Colwyn Bay all compete in England are historical although there are some grey areas for the non-league teams. Basically when the league was set-up in 1992 all Welsh clubs were playing in the English system of which the 'Welsh football league' was a division in non-league. Most Welsh clubs played in that, but those higher up, such as Bangor, Rhyl, Caernarfon, Newtown and others, as well as the ones stated above played in the higher echelons of the non-league and league pyramid against English teams.
In 1992 the FAW said that the non-league teams were told to join the newly formed national league. Eight of these resisted, including those listed above as well as Newport, Colwyn Bay and Merthyr. The league teams were given a period of five years' grace after which they too would join.
Only Merthyr were initially successful; the others then had long-running legal battles, which Newport, Colwyn Bay and Caernarfon won, while the others simply gave up and joined the league (although Caernarfon elected to quit the UniBond League a few years later and join the LOW).
The three league teams have never been approached legally despite the whole 5 years' grace having passed, so we still have the right to play in England and I suggest that if the FAW did try and force us to play in Wales we would resist strongly. However we are still full members of the FAW and are not registered with the FA. Technically, the FA wouldn't actually be able to nominate Cardiff for Europe even if they won the FA Cup because they don't have authority over them.
Also I find your comments about the Welsh Premier League to be pretty condescending and even factually wrong; there are actually two full time teams in the Welsh Premier, as has been mentioned.
As for the whole TNS thing, they are a club formed from a merger of two FAW-administered teams, Llansatffraid and Oswestry Town. Although Oswestry is actually something like 3 miles inside England they have always played their football in Wales so in light of that UEFA decided to let them keep playing here and represent us in Europe. That said, I do think we should be allowed to play in Europe by the same token. But it's not a situation that's likely to come up on any regular basis.