I think rather than use the word "important" a better choice would be "relevant". The past 200 years or so have more relevance to us as they have had a more recent effect on what has shaped our world as we currently know it. The accomplishments of 4000 to 5000 years ago have affected us, to be sure, but the events of that long ago have a smaller effect on our current world than have the past 4 or 5 centuries.
This is very true, however I would go a step further. What happened in ancient times is simply taken for granted in the modern day--the existence of Christianity, for example, or the Battle of Hastings (the one naval invasion of the British Isles, after which no one tried until Napoleon and Hitler centuries later), had massive effects on the modern world. It's just that people don't think about it. They see all the WW2 shows on TV and assume that the greatest military minds of all time faced off 60 years ago, and nothing in the world is any more important than that.
Byzantium?! They were an empire for 1,000 years, and covered quite a large area. This is much more of an accomplishment than early Greece managed. If anything, ditch early Greece.
I agree with the Celts.
HRE was more than just Germans, Charlemagne's HRE was mainly France.
Native America/Iroqois/Soiux Idians... AGREE completely. They were no empire, and they were rolled over pretty quickly. The Aztecs/Mayans were also rolled over, but there was at least a system there. N American indians were sparsely populated and except for a few rare occasions, never even united for a battle.
Portugal wasn't that big in Europe, but they got Eastern S. America, Africa and lands all over Indian/Pacific Oceans. That qualifies in my book.
The Dutch are a little questionable, but I love playing them... but its true... they did have some lands in Africa and in the Indian/Pacific Islands, just not nearly as much as Portugal.
Anyhow, long story short... ax any of the N. American Indian tribes. Tribes do not qualify as civilizations, particularly if there was no tech advance there AT ALL. Hunter/Gather societies...
Depends on how you classify the Byzantines. Strictly speaking, they are an offshoot of the Roman Empire. Then, they become a Greek Empire (adopt the Greek as official language instead of Latin) that is wholly unique from the classical Greek roots. Also, see the Persian Wars and the Greeks allowing a unique Western civilization to develop. Also, see Alexander's conquests. Also, see the contributions to science, medicine, astronomy, etc. that the Hellenistic cultures contributed to the modern world.
All that being said, I can survive with having Rome, Greece, and Byzantium all as independent civs.
On the native Americans, I'm not saying eliminate native societies, just replace the NAE with a specific tribe. The reason why they got steamrolled was biology--their immune systems were not up to the challenge of fighting European diseases. But don't mistake this for a lack of sophistication: the Aztecs and Mayans built aqueducts and pyramids in much rougher terrain than the Romans and Egyptians had to deal with. They had complicated social hierarchies and in some ways were more advanced than their European counterparts (look at the Iroquois on women's rights, for example). The Iroquois also had a functional representative government as well as good agriculture.
I'm not sure why you are talking about the Mayans getting rolled over. The Mayans disappeared before Columbus and the conquistadors arrived...you must be talking about the Aztecs and Incas.
Many mistake the lack of technological sophistication for a lack of social/economic/political sophistication. However, technology is not the sole requirement for being in Civ.
And I just can't believe you suggested the Dutch were questionable. Amount of land colonized is not what the Dutch are there for--they came up with modern banking, stock markets, and not to mention the contribution to painting, music, and the arts (the culture category of Civ4). And that's just off the top of my head, I'm certain there is more.