What does it mean to be integrated in society?
When sociologists talk about integration they refer to either A) integrating institutions, such as schools and religious institutions or B) the degree to which agents can participate in social practices, such as economic practice, cultural practice, political practice, etc.
If one dresses themselves in the attire common to that society but leads a mostly solitary or an alternative lifestyle, does that make them integrated?
Perhaps, it depends. The issue of social integration, from a sociological perspective, however has less to do with isolated individual instances, and more with social groups, e.g. latinos integrating with the American economy, the issue of social integration of African-Americans living in the ghetto, etc.
Should one be required to participate in social functions, adhere to social conventions or hold certain values deemed the "norm" by society or the state? Most of us live in an individualistic society where we can do our own thing as long as it is not illegal.
To be sure, this is a political question, not necessarily a sociological question. This depends on what one is talking about. States, for instance, do have certain requirements of participation, such as paying taxes. These are objective and legal requirements. There are other tacit and informal requirements: e.g. it is not required that you not be a Muslim, but if you where Muslim garb at an airport you may get suspicious looks and even questioned.
"Individualism" or the idea that we achieve our own destiny, craft our own lives and lifestyles according to taste is more an ideological belief than a reality. It takes social cohesion and institutions to achieve such a belief system and social structure to make it a reality.
If one does not follow the norms of society but is not damaging to others, are they undermining social cohesion and making the country weaker?
We have to be careful not to take "Society" as a thing in itself. Society is built up of a multitude of social practices within a particular social and regional context that overlap in some specific traits. It is by no means monolithic, especially in most of the contemporary globalized world. There can be much conflict within society, or latent conflict, and it still be "cohesive".
Truth lives, in fact, for the most part on a credit system. Our thoughts and beliefs 'pass,' so long as nothing challenges them, just as banknotes pass so long as nobody refuses them.
--William James