I really suggest you to use FlashDevelop as your default IDE.
This program is a lot lighter than Flash Builder (which is built on Eclipse).
If FlashDevelop is properly set up (a lot more simple than it seems), you can develop Flash-based applications completely free!
A useful shortcut is available too in FlashDevelop to compile your project in Flash IDE. Simple as [CTRL ENTER].
About your question about being a Flash developer and doing only programming, the answer is simple.
It's up to you.
I am presently working as a Flash AS3 programmer for a video game developer… and I am quite good at graphic design!
As I am working on a big project, I do not design at all because artists are hired to do this job specifically. But in another hand, if you're working in a small studio or if you are a student/hobbyist, you have more chances to draw/code.
Programmers do not design because normally they are not good at it.
But it doesn't mean that you cannot do it.
And yeah, we are using FlashDevelop at work!