Are there apps that enable "sticky" / "magnetic" / "docking" windows? I am looking for an app that makes windows stick to each other (if anybody used WinAmp back in the day you know what I mean).

This would be great when using apps that have several windows that I want to treat as one window - so if there is another method to accomplish this please do tell!

I found Zooom/2 that makes windows snap to each other when moving via Zooom/2's keyboard shortcuts, but the windows do not stick to one another when you move one of them.
Divvy was my tool of choice for a while. It works like Zooom/2, so it will not fulfill your needs completely, just as the other suggestions on this question.

Stata 10 used to have 'magnetic' windows but the functionality disappeared, much to my sadness, in Stata 11. This change makes me believe that OS X must be resisting that functionality.

I think that the whole Exposé logic is working against that kind of functionality in the minds of many (but not mine: I am very sympathetic to your problem, I hate messy screens too).