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you don't need R to be Euclidean. the claim is true in any domain because in every domain all non-zero elements have the same additive order, which is the charateristic of the domain.
for the second question just note that a group homomorphism $\displaystyle f$ is completely determined by $\displaystyle f(1)=\sigma,$ where $\displaystyle \sigma$ is any element of $\displaystyle A_5$ whose order is a divisor of 12. so we must have
$\displaystyle o(\sigma) \in \{1,2,3,4,6,12 \}.$ but $\displaystyle A_5$ has no element of order 4, 6 or 12 and it has 20 elements of order 3 and 15 elements of order 2. so there are 36 group homomorphisms from $\displaystyle \mathbb{Z}_{12}$ to $\displaystyle A_5.$