Basically the main idea behind the words with "делить" is division.

Here are some of the concept meanings of the words:


Делить - I take something and divide it into parts.

Разделять - I do the делить and then treat the pieces as standalone things.
(Same as "I can understand your oppinion = Я разделяю твоё мнение = I divide your opinion and take a part of it because I feel comfortable with it")

Отделять - I cut off (divide) a part of something, grab it and forget about the rest.

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Делиться - I do the делить to myself. That's the way an amoeba behaves.
Another one is that I do the делить to something I have and then give one or more of the parts to someone.

Разделяться - Something does the делить to itself and forgets that there was a time when it was a whole.
This happens when a group divides into smaller groups. Compare with разделять.

Отделяться - I do the делить and associate myself with one of the parts. That happens when I отделяюсь from a group of people. Compare with отделять.