For question #1, see this sequence, which contains all values $c$ such that $\mathbb{Z}[c]$ \mathbb{Z}[\sqrt{c}]$ is Euclidean: http://oeis.org/A048981
Note that these are the only such quadratic fields which are Euclidean.
For question #3, it is not solved completely since it is not known which values of $c>0$ produce a unique factorization domain. However, the problem is solved for $c<0$ completely via the Stark-Heegner theorem. The problem in general dates back to Gauss, and is known as the class number problem. Wikipedia has good info.
Using the answers to these, we can answer #2 since there are only a finite number of Euclidean quadratic fields.
