1) In neuroscience there's an enormous lack of enthusiasm for software development, even if it makes large parts of a workflow easier. If a poorly written, undocumented piece of code stuck in a file called temp123_final_final_no_really_final.m produces the same output as a higher quality piece of code, no one cares as long as it leads to a publication.
2) You don't need a degree in neuroscience to do neuroscience research, however you'll probably need to take a few basic neuroscience courses to get up to speed depending on what area you're interested in focusing on. It sounds like you're interested in theoretical neuroscience. For that you'll need to take lots of probability and statistics.
3) Make no mistake, your ability to write clean, well documented code will be massively under-appreciated by nearly everyone.
4) As far as division of labor goes, that usually depends on the lab size. The smaller the lab, the more you do. Even in some large labs you have people building software AND doing animal surgeries AND running experiments. There usually isn't a person dedicated to a specific type of labor.
5) To "fiddle" with the brain, you'd have to get into animal research as you can't really do that with people (yet?)
A lab worth checking out is the lab of Gyorgy Buszaki: http://www.buzsakilab.com/