Glad to help out. However, please be sure to read the note at the top of these forums that discusses writing out full questions. This is copyrighted material and shouldn't be copied in full.
That being said, this game asks you to determine a five-digit code. Each digit from 0 to 4 will be used in the code exactly once.
By the third rule, the second digit will be double the first digit. There are only two possibilities for that: The first digit is 1 and the second digit is 2; or the first digit is 2 and the second digit is 4:
1 2 _ _ _ or 2 4 _ _ _
The last rule tells us that the third digit will be smaller than the fifth digit. Write that down and keep it in mind throughout the game. That means the third digit can't be 4 (since that's the largest digit in the game) and the fifth digit can't be 0 (since that's the smallest).
For Q. 1, if the last digit is 1, then the first two numbers could only be 2 and 4. Furthermore, to satisfy the last rule, the third digit has to be 0. That leaves 3 to be the fourth digit:
2 4 0 3 1.
That makes (A) correct.
For Q. 2, (C) is correct because 2 has to be the first digit or the second digit (in which case the first digit is 1, meaning 3 would have to be later in the code.)
For Q. 3, we have to look at both possibilities.
For the first possibility (1 2 _ _ _), if the third digit isn't 0, it must be 3 (since it can't be 4 -- having to be smaller than the fifth digit). That makes the fifth digit must be 4, making the fourth digit 0. For the second possibility (2 4 _ _ _), the third digit would have to be1, making the fifth digit 3 and the fourth digit 0:
1 2 3 0 4 or 2 4 1 0 3.
In either case, (C) is correct.
For Q. 4, (E) is correct because if the third and fourth digits were 3 and 4, there'd be no digit left that would be higher than 3 for the fifth digit.
For Q. 5, (E) must be true. In the first possibility, 2 is the second digit, so 4 couldn't be any farther away than the fifth digit, which would put two digits in between 2 and 4. And in the second possibility, 2 and 4 are next to each other. So there can't be any more than two digits in between 2 and 4.