EU legislation doesn't use the word 'country' anywhere, and neither does anyone working in the Commission. The EU is made up of 'Member States' in the parlance.
Good thing too. 'Country' is a loaded word with various populations both claiming and denying their geographic locale as a separate country from the legal entity (or 'Member State') they're a part of.
They don't control the rest of Europe (though right now with the Euro situation they probably just about do), however there are provisions for legal precedent in one Member State to apply in every. I have no idea how it works and when it applies, though. It's definitely not so direct as in the U.S.
Courts in Europe are not precedence-based. At least not in all countries. Here (Poland) there is no such thing as precedence in courts, not to mention basing a verdict on precedence in foreign court.
- a state with lower-case S is a subdivision of a Federal Country.
- a State with upper-case S is a synonym of Country.
- All members of the EU are States/Countries. Some of these countries are Federal (Germany for instance) and are subdivised into states (Länder in the case of Germany).
- Some States of the EU include multiple Nations. UK is one of these. Scotland is neither a State nor a state for example, but it's a nation.
That's an amusing taxonomy, but don't fool yourself into thinking this stuff can be so precisely and unambiguously delineated. There's a continuum between a province with some autonomy to fully-fledged independent nation state, and the application of various English words to refer to particular entities that lie in different places on the continuum is governed primarily by history and culture, not by legal and political specifics. For any set of properties you care to define for nationhood or statehood, you can usually find entities that are not evenly cut into one or the other.
The Commonwealth shares a head of state; for example, Canada would be a sovereign state (small s) within the terminology of the Commonwealth. The Common Travel Area has passport-free travel; Irish and UK citizens can vote in one another's national parliaments, while EU citizens generally can only vote in local and EU elections. British crown dependencies are distinct from its overseas territories. There aren't enough shades of meaning in the miserly handful of terms you defined above (and seems rather German-centric) to adequately cover all the dependent and independent relationships involved.
I was merely trying to disambiguate the notion of state (as in United States of America) and the notion of State (as in Member State of the European Union), but you're right in putting a big warning sign on my taxonomy, since it's easily the kind of things that could (and actually has) lead to wars.
Oh and good thing my terms are German-centric, since I'm French. We don't even have the notion of state here in our civic system (but we do have tons of others of course).
"To vote in a UK general election a person must be registered to vote and also [...] be a British citizen, a qualifying Commonwealth citizen or a citizen of the Republic of Ireland"
The article doesn't use the word state anywhere, but one of the definitions of state is "a politically unified people occupying a definite territory; nation.", so yes, Europe consists of multiples states.