How do I add a field in the toolbar?
The examples module kinda does this:

 

it's not a drop down, but lists the links under. Take a look at // Create the examples toolbar render array. in examples.module to see how they do it. 



Now, I don't know if you can put an actual field in the toolbar with PHP code. The only thing that I can think of is to use CSS. I tested this and it worked. 

.field-in-question { position: fixed; top: 10px; z-index: 999; }