>Are they all built in South Korea (Hyundai, Daewoo, Samsung)?
So far, yes. At Hyundau HI, Hyundai Samho, STX, DSME, and Samsung. Orders at Hanjin (Philippines) have been canceled, but large container ships with a twin-island arrangement will also be delivered from Jiangsu New Yangzijiang, NAKS and plly a few others (CSBC) in the future. Basically all Japanse and Chinse Yards (i.e. Sadri) have similar designs ready. Furthemore, the design twin-isle principle will be applied to 9,000 teu ships too.
>How many different types (or sub-types) are there?
There is no answer to this since it depends upon when you consider ships identical: There are series of such ships with similar hull and same main engine, but slightly different superstructure and auxiliaries. At least a dozen variants exist.
>Do they differ much in size or capacity?
Presently between just under 13,000 (19-wide) and just over 14,000 (20-wide), including future deliveries the "type" (it's not a type actually...) will range from ca 9,000 teu (19-wide) to 16,000 teu (21-wide) and 18,000 teu (23-wide)
>Is there a common name for the "type" (like "Maersk E-class")?
No. Too many ships with too many idiosyncrasies will follow this arrangement, There is no point in lumping them together.
Edit: Hyundai MM has to be added to the list of operators which already have such vessels.