There is a miscalculation since Vanilla on Quick speed, where the culture required for Legendary level (and 80%-level, IIRC) is counted not as 2/3 of Normal (2/3*50000 = 33333), but as 1/2 (25000). Therefore it is almost 20% cheaper to achieve Legendary status on Quick.

Also, units on Marathon (except Settlers) cost 2* of their Normal cost instead of 3*.

I wouldn't consider either of those miscalculations because they are explicit XML settings; there's nothing for the game itself to calculate. The Legendary culture values are from CIV4CultureLevelInfo.xml (and every one of the Quick culture values is 1/2 Normal) and the Marathon Unit costs are from CIV4GameSpeedInfo.xml

While you can certainly argue that they are inconsistent and so could be mistakes, given the way they are implemented and how long they've been that way, I don't see them as bugs but rather design decisions that I'm hesitant to mess with. It would also be trivial for a user to change them to suit his/her needs.

Ok, fair point

The unit production definitely is when you consider on marathon, units move faster so it is more war oriented.. and so unit production is somewhat between movement (1x) and nonmilitary production (3x)

As for the Cutural... not sure, but I'd leave it as is

Here's a possible reason why it's less than 2/3 requirement for quick speed specifically.

You're probably aware that buildings which produce culture double their cultural output after 1000 years. If you looked at the way years match turns for quick speed, you'll find a larger proportion of the game is spent after about 1900AD. Since this doubling of culture is before the multipliers you'd typically have in your culture cities, they may have decided this accounts for a good few thousand culture.

Personally I don't think it would account for that much but there may be other factors similar to this which makes quick speed a bit more unique.

Here's a possible reason why it's less than 2/3 requirement for quick speed specifically.

You're probably aware that buildings which produce culture double their cultural output after 1000 years. If you looked at the way years match turns for quick speed, you'll find a larger proportion of the game is spent after about 1900AD. Since this doubling of culture is before the multipliers you'd typically have in your culture cities, they may have decided this accounts for a good few thousand culture.

Personally I don't think it would account for that much but there may be other factors similar to this which makes quick speed a bit more unique.Well, the cost of buildings is the same, and if the wish was of levelling out dates, it would've best been done directly by aligning the date-turn relation. The compensation for higher unit logistics costs and little rushing option is the most probable reason, IMO.

The unit production definitely is when you consider on marathon, units move faster so it is more war oriented.. and so unit production is somewhat between movement (1x) and nonmilitary production (3x)

As for the Cutural... not sure, but I'd leave it as is
I suppose it was made to make things more lively to the player - for regular 2/1/1 capital and a wood building first worker for 180/4 = 45 turns would've been excruciating.