"this" refers to something close, "that" refers to something further away.
der, die, das basically mean "that", but can sometimes mean "this" as well; usually, both translates are accepted.
dieser, diese, dieses; dies generally only mean "this".
Note that "nobody" is grammatically singular in English, so it would have to be "nobody likes", not "nobody like".
Is the phrase "Jeder nicht mag das" something people would say in German
No. It's grammatically incorrect -- the verb is not in the second position in the sentence where it belongs.
"Everybody dislikes that"
German doesn't have a commonly-used word for "dislike", so you would instead say "Nobody likes that": Niemand mag das.