Incorrect.
Why? "It's" is a contraction, meaning "it is". "Its" is a possessive (shows ownership).
The sentence should be: The dog picked up its bone.
Correct.
Why? Use a comma to separate each item in a series, including before the conjunction (and).
Incorrect.
Why? The word "got" should be used and "gotten" should not be used.
The sentence should be: The kids returned without having got any candy.
Incorrect.
Why? "Less" refers to a quantity or amount. "Fewer" refers to a number. For example, you can say: We had less trouble in math class today because there were fewer students.
The sentence should be: They had fewer students in math today.
Correct.
Why? When talking about yourself as the subject, use "I" and put yourself after the other person in the subject. When talking about yourself as the object, use "me". For example, you can say: My friend is coming home on the bus with my brother and me. (not "I")
Incorrect.
Why? This is a double negative (don't, no). The "don't" cancels out the "no", meaning you do have homework.
The sentence should be: I don't have any homework or I have no homework.
Source: Strunk, W. & White, E.B. (2000). The Elements of Style (4th edition). NY: Longman.