I won't do these problems "thoroughly" but only remind you some very basic facts since, as you say, you already know the elementary stuff around this
subject: take problem (1) for instance, and let be a ring homom., then must divide 30 AND ALSO 12(for example, because
any such ring homom. is also a group homom. and thus by Lagrange's theorem).
Now, since as an additive group (meaning: every element of the ring is of the form for some and, in fact, we can choose since this
is just the ring of residues modulo 12), then is uniquely and completely determined by its value on , so you've to count all the possible choices for the
image and s.t. the subgroup has order dividing 30 and 12...
For example, if as an additive group, then it CAN NOT BE that , since both generate the whole , but is fine since ....etc.