In your first expression

In our notation, those e's are sometimes just positive numbers, and other times they are the identity element, but it doesn't define it here. If it is the identity element, could someone elaborate where they come into play?

all of the P s are prime, so the e s are the powers of each prime in the prime factorisation of n.n=(2^e(0))*(P(1)^e(1))*....*(P(k)^e(k))

yes

For an example, is it correct to say:

if n=pq where p and q are distinct primes, and (a/n)=-1,

then (a/n)=(a/p)*(a/q)=-1?

If you know latex, you can access the displaymath enviroment by entering or clicking the button with on it. If you don't know latex, you can how other people made pretty equations by clicking on them.Sorry for the cramped notation, I've seen people use the proper notation on these forums, but I don't know how to do it