Hi,
1) Let's call it a day => I think it means "Let have a day off", but I can only guesss here No. It's a common expression that means 'We have done enough work for today, we have completed our day.'
2) alligator and crocodile, => they are the same meaning...but why do they call each other with the animal name? It's a casual, humorous way for friends to say good-bye to each other. It was originally made popular years ago in a silly song, which included the words:
See you later, alligator 'See you later' rhymnes with 'alligator'.
In a while, crocodile 'In a while ' rhymnes with 'crocodile'.
I think the writer chose these two reptiles mainly because of the rhymning, but also because they are so similar that you could kind of imagine them having a conversation. It's just a silly song.
3) after a while => I think it means "see you later" Yes. The phrase in the song is 'In a while', same meaning.
See you later, alligator! Clive