I searched the literature and found that the cyclic groups are completely classified. If one restricts one's attention to rings with torsion-free additive groups, then the answer is know fairly precisely (all ingredients are known, but the classification of recipes is incomplete). I include a different kind of result on division rings, as it is used in the torsion-free case and is interesting at any rate.
I suspect the literal answers to all of your questions is that we do not know.
Cyclic groups
Pearson–Schneider (1970) classify those cyclic groups that occur as groups of units of rings. They are exactly the infinite cyclic group and the finite cyclic groups whose order can be written as a coprime product of numbers of the following forms (forms may be repeated):
- $q^t-1$, $q$ prime, $t \geq 1$
- $q^s(q-1)$, $q$ and odd prime, $s \geq 1$;
- $4m+2$, $m \geq 0$
- $4n$, $n$ odd positive and if a prime $p$ divides $n$, then $1 \equiv p \mod 4$.
The paper is:
Division rings
Amitsur (1955) classified the finite subgroups of division rings. Note that the group of units of a non-commutative division ring is never itself finite, so this is not directly relevant to the question (but I suspect is interesting to readers).
- Amitsur, S. A.
“Finite subgroups of division rings.”
Trans. Amer. Math. Soc. 80 (1955), 361–386.
MR74393
DOI:10.2307/1992994
Torsion-free rings
Amitsur's work was generalized to rings whose additive groups are torsion-free by Hirsch–Zassenhaus (1966). Here the groups must be built in a simple way from just a few small groups: C2, C4, C6, Q8, BT24, and DC12. The automorphism group of the additive group is the group of units of the endomorphism ring of the additive group, and conversely Corner has showed that every countable ring with a torsion-free reduced additive group is an endomorphism ring of a torsion-free abelian group. Hence the finite groups involved are the same.
Corner, A. L. S.
“Every countable reduced torsion-free ring is an endomorphism ring.”
Proc. London Math. Soc. (s3) 13 (1963) 687–710.
MR153743
DOI:10.1112/plms/s3-13.1.687
Hirsch, K. A.; Zassenhaus, H.
“Finite automorphism groups of torsion-free groups.”
J. London Math. Soc. 41 (1966) 545–549.
MR197549
DOI:10.1112/jlms/s1-41.1.545
Best Answer
A quick google search finds the paper Finite Rings Having a Cyclic Multiplicative Group of Units by Robert W. Gilmer, Jr. which settles the question as follows:
This is complemented by proving that every finite commutative ring is a direct sum of primary rings and that its group of units is the direct product of the group of units of the primary rings, and is cyclic iff each part is cyclic and they have coprime orders.