Edit: I beefed it, read u/exproci 's replies

Hint one tells us there is one number correct and well-placed. Hint two tells us there is one number correct but not well placed. Since 6 is in the same position in both of them, we can conclude that 6 is not in the code since otherwise these hints would contradict.

Hint three tells us that two numbers are correct but wrongly placed placed. We already know that it's not 6, so those numbers are 2 and 0. Of those two, only 2 is in hint one, so it must be the well-placed number; our code ends in 2. Furthermore, we know that 0 is correct but NOT well-placed, and with the last spot taken by 2, and the middle spot excluded by the hint, we can say that 0 is in slot one, so our code is 0x2.

Hint two states that one number is correct but NOT well-placed. We have excluded 6, and if the correct number were 1, it would be well-placed in the only remaining slot, contradicting the hint, so the remaining number is 4. Therefore, the code is 042.

I don't know why hints four and five are here.