I understand why these statements may appear to be contradictory, but - as with many things - much of the confusion probably stems from the terminology being used. So, firstly, let's be clear about definitions.
The Ancient Egyptian civilisation is generally accepted to have originated in the Neolithic with the Faiyum A culture in Lower Egypt (with evidence dating back to about 6000 BCE), and the Badari culture, which has given us the earliest evidence of agriculture and permanent settlement in Upper Egypt dating to around 5000 BCE. These were followed by the more-familiar Naqada Neolithic cultures, also located in Upper Egypt.
Upper Egypt is broadly defined as the Nile valley between the Cataracts of the Nile (above modern-day Aswan), downriver (northwards), roughly to the area of modern El-Ayait.
The Ancient Egyptian name for this region was written in hieroglyphs as:
which transliterates as tA Smaw, and might be translated - rather appropriately - as "the Land of Reeds".
[Lower Egypt] is, broadly speaking, the Nile delta. It runs from modern El-Ayait north to the Mediterranean.
The Ancient Egyptian name for this region was written in hieroglyphs as:
which transliterates as tA mHw. Literally, "the land of papyrus".
Just to complicate matters, the northern part of Upper Egypt, between the city of Sohag and El-Ayait is also known as Middle Egypt
There were indeed two crowns associated with Upper and Lower Egypt. The White Crown, (ḥḏt) or Hedjet was the crown of Upper Egypt:
and the Red Crown, (dSrt) or Deshret was the crown of Lower Egypt:
After the unification, these crowns were combined to create the sḫm.ty or "Pschent" double-crown:
In the combined crown, sḫm.ty, the White Crown appears above the Red Crown, and I suspect that this is what you are referring to when you say that:
"... there is a crown of Upper and lower Egypt, with the southern or white crown (Upper Egypt) being superior to the red crown".
Although, as we'll see below, the unification of Egypt was achieved by the kings of Upper Egypt, who then became the kings of the new unified kingdom, so the ḥḏt crown might also be considered 'superior' in that sense.
The Origins of the Ancient Egyptians
This is actually one of the hot topics in modern Egyptology.
As I mentioned above, the earliest evidence we have for permanent 'settlement' and agriculture in Ancient Egypt is from the Neolithic Faiyum A culture. This dates from about 6000 BCE and located in the Faiyum basin in Lower Egypt.
The Faiyum A culture has also given us the earliest evidence for weaving in Ancient Egypt. Unlike the later Neolithic cultures of the Nile Valley, there is no evidence to suggest that the Faiyum A culture ever developed anything that we would recognise as a permanent village or town. The only permanent, fixed, features that we have been able to identify are hearths and granaries.
Nevertheless, if you want to define the origin of the Egyptian civilization as the earliest 'settled' culture, then the Faiyum A culture in Lower Egypt (i.e. the North of the country) would be a contender.
On the other hand, many - perhaps most - people would include permanent villages or towns among the hallmarks of a 'civilisation'. In that case, you would probably have to consider the Badari culture to be the origin. After all, this is the culture for which we have the earliest evidence (dating to around 5000 BCE) for agriculture and permanent settlement in Upper Egypt (i.e. the South of the country).
If you are looking more generally for the origins of the Ancient Egyptians as a people, the latest DNA analysis that I'm aware of (involving 166 samples from 151 mummified individuals), which provides:
"the first reliable data set obtained from ancient Egyptians using high-throughput DNA sequencing methods and assessing the authenticity of the retrieved ancient DNA via characteristic nucleotide misincorporation patterns and statistical contamination tests to ensure the ancient origin of [the] data."
found that the ancient Egyptians actually most closely resembled ancient and modern Near Eastern and European populations, especially those in the Levant.
Although only 3 complete DNA sequences were obtained (largely due to the difficulties of recovering uncontaminated ancient DNA), the team also recovered mitochondrial DNA from 90 of the individuals tested. The study included samples from individuals from Upper Egypt, Lower Egypt and the Faiyum. The DNA analysis did not indicate any significant differences between the origins of the peoples of any of the parts of Ancient Egypt.
This last finding agrees with the results of a 2007 study into craniometric variation, titled
Population Continuity or Population Change: Formation of the Ancient Egyptian State and published in the American Journal of Physical Anthropology, which indicated:
... overall population continuity over the Predynastic and early Dynastic, and high levels of genetic heterogeneity, thereby suggesting that state formation occurred as a mainly indigenous process.
At this point, it is probably worth mentioning the oft-quoted (or, more accurately, 'oft-misquoted') study of the DNA of the Pharaoh Ramesses III. Reports in the popular press have frequently claimed that he was of "African descent" because the analysis showed the presence of the E1b1b haplogroup.
The analysis was carried out by the company, DNA Tribes, and their findings were reported in the the DNA Tribes® Digest February 1, 2013. Their conclusion actually states:
These results indicate that both Ramesses III and Unknown Man E (possibly his son Pentawer) shared an ancestral component with present day populations of Sub-Saharan Africa. This preliminary analysis based on eight STR markers does not identify the percentages of Sub-Saharan African ancestry for these ancient individuals. This preliminary analysis also does not exclude additional ancestral components (such as Near Eastern or Mediterranean related components) for these ancient pharaonic Egyptians.
In addition, these DNA match results in present day world regions might in part express population changes in Africa after the time of Ramesses III. In particular, DNA matches in present day populations of Southern Africa and the African Great Lakes might to some degree reflect genetic links with ancient populations (formerly living closer to New Kingdom Egypt) that have expanded southwards in the Nilotic and Bantu migrations of the past 3,000 years.
This is entirely consistent with the results of the study by the Max Planck Institute, discussed above.
[In this context, it is also worth mentioning that haplogroup E-M2 (formerly referred to as E1b1a) is believed, on present evidence, to have originated in the Horn of Africa around 42,000 years BP. Frankly, given the proximity to the Nile Valley, it would indeed be remarkable if this haplogroup were not represented in Ancient Egyptian populations].
However, rather than describing the Ancient Egyptians as "middle Easterners" or a "Semitic people" (neither of which are correct), on the basis of the best currently available evidence, it is perhaps better to think of them in their contemporary context as simply an "Eastern Mediterranean people".
The Unification of Egypt
We have good archaeological evidence for a number of conflicts in the Predynastic Period in Ancient Egypt.
These initially resulted in the emergence of the two 'kingdoms' of Upper and Lower Egypt (tA Smaw and tA mHw). These two kingdoms were eventually unified into a single kingdom with a single ruler. The evidence at present is incomplete, but the process of unification seems to have taken a number of years, and appears to have been completed during the reign of King Narmer. This is probably the event famously commemorated in one of the best-known artefacts from Ancient Egypt, the Narmer Palette:
Since Narmer was the king of Upper Egypt (i.e. the South of Egypt) before unification, this meant that the first kings of the new unified kingdom were, indeed, 'from the south'.
The Location of Pathros
As regards the quotation from Ezekiel, I would generally be cautious when citing the Bible as an historical document. When considered in isolation, it is at best, an unreliable historical source. That is not to say that it is entirely without value as an source (and it is most certainly not "just a work of fiction"!). I would recommend taking a look at Converting the Past: Studies in Ancient Israelite and Moabite Historiography, by K.A.D. Smelik (especially the first paper, titled 'The Use of the Hebrew Bible as a Historical Source: An Introduction') if you can get hold of a copy. Professor Smelik identifies many of the challenges we face when trying to use the Bible as an historical source and several of the ways in which these challenges might be overcome.
On the more general question of the identification of "Pathros" as "Upper Egypt", @LangLangC has already dealt with this in some detail in his answer.
However, I'd just like to add a link to a 1959 paper, Egypt and the Bible: Some Recent Advances, by the Egyptologist and Bible scholar, Kenneth Kitchen, which includes some additional observations on the etymology of the name "Pathros" for "Upper Egypt" and which you may find interesting.