Is There More to Discover about Four Consecutive Numbers?

This problem follows on from, Continuing to Explore Four Consecutive Numbers.

Take four consecutive numbers, $a$, $b$, $c$, $d$.
 

1. What is $(a^2 + d^2) - (b^2 + c^2)$ equal to?  Show that it always takes this value.
    
2. Why is $(a^2 + b^2 + c^2 + d^2) - (1+2+3)$ always divisible by $4$?
   Why is it always divisible by $8$?
    
3. Why is $abcd$ divisible by $24$?
    
4. Explore $\sqrt{abcd + 1}$.

With thanks to Don Steward, whose ideas formed the basis of this problem.