Handwaved answer to your first question is that the national identity as a Big Thing likely emerged as a result of a 30-year-war (more specifically, Peace of Westphalia), with nation-state being an efficient unit for maintaining balance of power that the Westphalian system demanded.
To quote from Wiki:
The edicts agreed upon during the signing of the Peace of Westphalia were instrumental in laying the foundations for what are even today considered the basic tenets of the sovereign nation-state.
Aside from establishing fixed territorial boundaries for many of the countries involved in the ordeal (as well as for the newer ones created afterwards), the Peace of Westphalia changed the relationship of subjects to their rulers. In earlier times, people had tended to have overlapping political and religious loyalties. Now, it was agreed that the citizenry of a respective nation were subjected first and foremost to the laws and whims of their own respective government rather than to those of neighboring powers, be they religious or secular.
Why has national identity or being part of a country, become so important, as opposed to being part of a neighborhood, a city or the whole world?
As far as why not whole world, it's not an easy question. The best answer probably lies in the fact that anything beyond a nation usually has has too divergent interests, AND is too large to effectively govern long term absent a strong central tyrant (see Alexander's empire, or Ghengizids).
You can see it happening in EU issues in 2011 - Greeks and Germans just can't exist as a single unit, since either one culture will feel as if they are being taken advantage of (Germans don't want to pay for "lazy not-working-enough Greeks" social spending), or some strong central power - which doesn't exist in Europe - must mandate and enforce uniform rules.
As far as "why not city/neighborhood" - because the modern system of the world, both military and industrial and economy, makes things much more efficient when unification synergies are realized, and make the world uniform enough (due to literacy/printing press, among other things - thus Peace of Westphalia as a start) that uniting many cities into a nation doesn't present THAT much of a conflict of interest.
Answer to your second question is "absolutely yes". As a random example, plenty of immigrants from USSR to USA feel 100% American and zero percent either Soviet or Russian. If you want a reference to specific example, you are reading his writing this second.