This equivalence is false. Indeed, consider n = 4. Then there exist r = 17 and s = 999 such that n = r * s implies r = 1 or s = 1. The implication is true because the premise is false. Thus, the right-hand side of the equivalence is true while the left-hand side is false.
This is also false; consider n = 5 and r and s as above. The statement should be, "n is composite <=> and and .
This is false for all n because for r = n and s = 1, the premise of the implication in the right-hand side is true, but the conclusion is false.
The idea is that n is prime if all its divisors are trivial, but n is composite if there exist nontrivial divisors.
Also, universal quantification is usually used with implication: while existential quantification is usually used with conjunction: (though this is not an absolute rule, obviously).