Just who has final say over EU affairs? Is it the individual states or Brussels?

Answer 1

I think your very question demonstrates why the EU probably won't continue in it's present form....there's too much nationalism there.

Answer 2

It really depends on the topics but most of the time, it is the quad Germany, France, UK and Italy that indeed have the most influence.

European Union was built with compromise as value, so it is normal that there are discussions between states before a decision becomes European.

So to answer to your question, I believe Brussels (=European Commission, EU Concil and then EU Parliament) decides for EU affairs, after having consulted its 27 member countries.