Identifying the configuration of transisitors
My intuition says the base needs to have some power source.


At first glance it does look that way, however, you have to remember that ground is really just another voltage level. If you look more closely at your circuit you will see that..

Since the base is connected to a higher voltage than the emitter.. the base DOES have a power source!


  My question lies in identifying the configuration of the transistor [Common: {Base,Emitter,Collector}]


Does it matter for this problem? 

In reality, since there is no indication of an output in the schematic, or the question parts, it is not possible to tell which one it is. The best you can state is "It is probably not common base."

But it is not relevant. 

The math required to answer the problem parts is the same regardless.