Random Puzzles 

Workings

Hint
Start with the possible answers where the first digit is a prime number, and then look at the third digit.

Answer
23,475.

Reasoning
By (1), the first digit is prime:
   2----
   3----
   5----
   7----

By (3), the third digit is twice the first digit, so we can eliminate 5---- and 7----:
   2-4--
   3-6--

By (4), the fourth digit is the third digit plus three:
   2-47-
   3-69-

By (5) the fifth digit is the difference between the first digit and the fourth digit:
   2-475
   3-696

We know from the introduction that no digit is repeated, so we can eliminate 3-696. And, by (2) the second digit is the fifth digit minus the first digit:
   23475

Workings

Hint
Look at Billie's statements first.

Answer
Charlie committed the terrible crime.

Reasoning
Looking at Billie's statements, one of the statements was a lie and the other was the truth. Therefore, it must have been either Charlie or Emery.

Looking at Dale's statements, for the same reason, it was either Charlie or Billie.

Therefore, it must have been Charlie who committed the crime.

Double-checking this against the other statements confirms this.

Workings

Hint
Look at the initial letters.

Answer
Elephant.

The initial letters form the word ECLIPSE, so any animal beginning with E will work.

Workings

Hint
A little self promotion.

Answer
a.

From BrainBashers, the title of this website!