You lose a great deal of your identity as a person with a specific tribe that has a specific history and a story to tell. There are also issues that pertain to sterility.
Identity is only what you make of it. How you choose to identify yourself can often rely on upbringing and what makes you happy. I call myself English, but I am actually part Irish and was born in Wales. Are they a part of my identity? Hell yes. I love the fact my nan's Irish, I may not have grown in Ireland, but to my mind being part Irish just adds to the wealth of my history and the stories told. The English history in itself is incredibly wealthy and my identity as an Englishman is made up of several nationalities, not just Angles and Saxons. Many tribes mixed to make up the English race. Yes, we don't have Angles, Jutes, Saxons, Vikings, Normans, Celts and so on any more, but they do make up our national identity. We revel in its History. We learn about it from a young age and each stage of our History is told to us, the stories are passed on. Those of us who choose may explore parts of our History even more. For me, I have a great interest in the dark ages. Saxons vs Vikings, the birth of the English language and Danelaw.
Granted my interest weighs more with the Vikings, but I think it's fair considering the part of England I live in (East Anglia) was a part of Danelaw and therefore was under the influence of Vikings. As somebody who's part Irish, part English there's a lot of stories I could tell you and a lot of history I can share.
Stories will only live if they are told and history will be remembered if it's taught. Your national identity ought not come into it.
If you're talking about more recent history, for example, family history, I don't think in our modern society there's as much storytelling as there could be, I would suggest that's down to technology rather than multi-culturalism and the fact we live in a literate society as opposed to an oral one. Storytelling was a very large part of an oral culture and hence stories would be passed down from generations of people with their own personal touch, but we a literate society we learn things in a whole different way.
But I know in the UK there's a lot of interest in family history, heck we've chased our own family history and learned some interesting things about our own relatives, like my great granddad being in the war with Spike Milligan or tracing relatives back to the Victorian era and finding them of certain nobility, it's also finding interest in one our family names, which is a very rare name and seeing how it has spread out. Then there's tracing the Greek blood, you can certainly see it in my great uncle (whose surname is still Greek).
The point, identity is only there if you choose to keep it. One can lose identity in a number of ways and if people wish to give it up, it's up to them. People will do what makes them happy.
So when people mix nationalities they don't have to lose culture or their own identity, they can take on the culture and identities of all their nationalities. If a person wishes to mix identities or even dispose of one, what's exactly the problem? People will live their lives in the ways that make them personally happy. It isn't necessarily a detriment to society, because society has always mixed nationalities, it is not a recent phenomenon.
If you're not a fan of mixing cultural identities and multiculturalism, then identify yourself by one nationality and uphold any traditions of your own countrymen. It would be a fascist act to force people to preserve their own identity in blood rather than through memory.
Sorry didn't have time to address your other points, but I'm sure I shall return.