Which model may be best for outcome of a surgery?
Question 1: I would propose Decision Trees as a first thing you should look into, as they are easy to implement and the results are straightforward to interpret. 
You will end up with a tree-structure, where the nodes hold intervals/values of your variables. At each node, the tree tries to split your data most efficiently into your binary classifications $C_1$ and $C_2$. Thus, in the end, you will be able to extract most important features in your variables at the top of the tree. A typical measure of "efficiency" when selecting attributes is Entropy. 

Question 2: I think it is also solved with the above method. For each node of the tree you know the amount of $C_1$ and $C_2$ classifications on the left and on the right. Assume your attribute in the node is "Has diabetes". Your data (and thus the tree) tells you, 100 out of 110 patients "failed" the operation if the person had diabetes. Thus, you can give an estimate about how certain attributed contribute to success of your operations, and conclude it would not make much sense to operate the given person. 

Nonetheless, always be careful about how you interpret the results. You should not try to interpret attributes that split small subgroups, since this can easily lead to false assumptions about real-world behaviors.