I agree with you about Saigon, but I'm not sure about New Delhi. I think it's fine to keep it the way you have it, since that's how your source has it, but it is written as the capital of India under the World Capitals Quiz. If you were to change it, you could add Dubai (4.63 mil) under D instead.
Of course, it's your quiz and your choice. I think New Delhi would be more consistent with the site overall, but Delhi would be consistent with your source.
Although colloquially Delhi and New Delhi are used interchangeably to refer to the National Capital Territory of Delhi (NCT), these are two distinct entities, with New Delhi forming a small part of the city of Delhi. The National Capital Region is a much larger entity comprising the entire NCT along with adjoining districts in neighbouring states, including Ghaziabad, Noida, Gurgaon and Faridabad.
You're definitely going to have explain how you came up with 7.31 million for Essen. The city of Essen is 582 415, although it's part of the Ruhr Area, which has an urban population of 5 118 681 and a metro population of 10 680 783. Assuming that the 7.31 million includes some, but not all of the people in the Ruhr Area, it might make sense to attribute all of these people to Essen if it was the largest city in the Ruhr Area... but it isn't; Dortmund is. If the whole area is going to be attributed to just one city, it should go under D for Dortmund.
I would also use Essen because Dortmund, Duisburg and Essen have about the same size while there are many cities (bochum, gelsenkirchen, mülheim etc.) that also have much more than 100k in poulation. The thing is that Dortmund and Duisburg are at the very edges of The Pott, while Essen is just in the center.
You easily get different population values for The Pott, because its borders are hard to define. For example Hamm could seen both as part of The Pott or not.
I've done a couple of these now and I think the time needs to be doubled. On my first go I had never heard of most of the cities, so more time would not have helped, but it would with some letters.
-New Delhi has 21.75 million people but is not on here
-Saigon has 8.99 million people but is also not on here
Saigon already change to Ho Chi Minh City after Vietnam War.
Of course, it's your quiz and your choice. I think New Delhi would be more consistent with the site overall, but Delhi would be consistent with your source.
You easily get different population values for The Pott, because its borders are hard to define. For example Hamm could seen both as part of The Pott or not.