The explanation I found here seems alright.
We use of with (a) little and (a) few when they come before articles (a/an, the), demonstratives (this, that), possessives (my, your) or pronouns (him, them):
Put the flour into a bowl, blend with a little of the milk, beat in the egg yolks, then the sugar and the rest of the milk.
A few of his films were seen abroad.
All the sentences except 2 are correct.
A little bit of bread is fine because here, little describes bit of bread. Of has nothing to do with little. You say, a bit of something and a little bit of something.
A little of bread sounds unnatural. You could say a little bread or a little bit of bread instead.