GetCommonBaseClass: Readabiliy and Functionality
I think that code A is quite a lot simpler, so it's also more readable. It's quite clear what GetCommonBaseClass(Type a, Type b) is supposed to do and GetCommonBaseClass(params Type[] types) is also quite straightforward.
On the other hand, CommonBaseClass(params Type[] types) in Code B is much more complicated, it's hard to see what's actually going on in all that code.

Also, in code A, GetCommonBaseClass(params Type[] types) could be simplified even more by using something like types.Aggregate(GetCommonBaseClass).

Regarding the edge cases, I think that, null inputs should cause an ArgumentNullException to be thrown. (Or not, if null is really a valid value.) With interfaces, I think the correct behavior depends on your requirements. If you don't know, I think the safest choice is to throw an exception. It's trivial to allow interfaces later on, if that's needed. On the other hand, adding a restriction could cause a issues with older code that uses your method.