They don't have their own seat at the UN for instance. They don't field their own team at the Olympics.

If Wales needs a tld, what about Texas? Texas has a distinct culture and was once an independent republic. I'm sure people from Wales think they are independent in some way that Texas is not, but people from Texas think those thoughts too.

It is actually extremely murky, because there is no single document which states what the constitutional status is, there is a patchwork of legislation which refers to England and Wales, then there are the Wales-specific acts such as the Welsh Language Acts, and then there's the devolution and the devolved legislation (which establishes that Wales is to some degree independent within the UK). The constitutional problem, in a nutshell, is that Henry VIII introduced the Laws in Wales Acts to create the legal entity "England and Wales", but there had been no legal entity "Wales" before he did.

Also, if Wales is already a country, why is independence the stated aim of Plaid Cymru :)?

The EU? The UN?

Wales is a constituent country of the United Kingdom and has some devolved power. However, it is the United Kingdom which is internationally recognised.