I'll give it a go... :)
Date: custom field (tag) you can fill out yourself, generally refers to the date a song came out or whatnot... with images you can just set it to the date the picture was taken (which would -usually- be the same at the other dates, unless you left the picture in the camera for a year :))... it seems this field is automatically filled in with the Date Created for images, when they are first imported.
Date Created/Modified: these are both OS controlled fields as opposed to tags.
Created: the date the file was created on your computer (or in the camera... don't know exactly when that data is filled out).
Modified: the last time that anything in the file changed (which would include tags for music, if they are being updated).
Date (day, month, year, filename friendly): I believe all of these are derived from the 'Date' field and filled in automatically from the 'Date Created' field for image files when they are first imported. You can change them yourself (excepting filename friendly, but if you change the others this'll update) if the date created isn't the date you want tagged... changing these updates the 'Date' field.
Date Imported: This is the date that the file was imported into Media Center.
Date (filename friendly): This is linked to the Date field (updates anytime you change the Date or Date (day,month or year) fields) and is readonly (can't be modifed directly by the user). I'm not sure what people use it for though... anyone?
2) I suspected that the filename friendly, month, day, and year were calculated from the Date, but they don't update if I change the Date manually? What is the relationship between these?
Changing the Date field changes the Day/Month fields and vice versa on my MC (as long as I type a valid date, otherwise Day/Month are just set to blank out)... what version are you using?