Does national identity depend on an independent currency?
Well, most countries have drawn national symbols in their coins and bills. It is a good way to feed the national identity and spread the myths, events and the personalities that forged the history of a country. In that sense, yes, an independent currency helps to sustain the idea that every person bearing a coin with the symbols of a community is a member of it. But, the inverse, ie, the question you asked is not true. One does not need to pay his things with coins and bills carrying the ensigns of the country one regards as his own, to feel a part of it. In many parts of the world you have or had numerous examples of nations that were part of some state that comprehended many other different national identities, all using the same currency, and that national identity never disappeared. National identities depend on many things and a currency of its own is not an indispensible condition to sustain that identity.
Does a country give up control of it's economy when it gives up control of it's currency?
In the case of Europe that is true at some extent. It's logical that when you have 12 countries that abandoned their national currencies to adopt a common one, the decisions concerning the monetary policy will have to be shared. And offcourse, this policy naturally affects the economy. The funny thing is that most countries gave up their monetary independence with so little dificulty, because they thought that it was, in fact, the only way to gain again some control of it. Before the Euro, all the euroland countries would have to follow what the Bundesbank was doing. France, for example, accepted this in order to have a word to say, instead of being completelly dependent on the decisions that were coming from Frankfurt. Now, they still come from Frankfurt, but the bankers from all the Euroland can take part on it. In exchange, France supported the full political independence of Germany.
Now, the big fight inside the ECB (European Central Bank) is about what type of guardianship it should follow. If it should be ruled by the central bankers and completelly independent of the political power (Wim Duisenberg is within this line). Or if, as the french want, it should remain more attentive on the political signs (like the Federal Reserve).