I've looked up a few dictionaries and a sentence database, and I would say that you are correct, at least in terms of modern English usage. Winnie the Pooh was written in the 1920s and it is possible that then "check off" meant "cross off", "delete from". If not, then you may have discovered a minor mistake made by A.A. Milne. You would say : "checked off each item on the list" or "checked each item on the list off", but not "checked each item off on the list". Note the position of "off". I prefer the first option because the second has too many words between "check" and "off". Good question.

Yes, either is fine. You can see example sentences without "on" here: http://www.oxforddictionaries.com/definition/english/check-something-off?q=check+off And one with "on" here: http://www.oxfordlearnersdictionaries.com/definition/english/check-off?q=check+off