Are live lessons usually about generally improving technique in the voice/what the voice teacher thinks should be improved or do teachers just help with whatever the student wants or is it a bit of both?

What generally happens in the first lesson? Is there some kind of test/analysis of where the student is vocally at the beginning or do you figure that out as you go along?

It's both! A teacher's job is to help you sing healthily and grow your skills, while also helping you reach your specific goals. 

In every lesson, we'll do some vocalizing (exercises) and song work (or ear training, if it's needed and/or within the student's goals). Analysis is continuous (at least for me - I know some teachers will do a "diagnostic" scale at the beginning, but exercises and song work serve as that for me).