I wrote a SIMBL plugin at one point to neuter applications that did this too much (actually, I forked it from someone who wrote it for Spotify). Trying to grab my attention in a persistent, repetitive way for something that is minor drives me up the wall.
Exactly. I found the bouncing animation genius, as it always reminded me of a toddler impatiently jumping up and down for attention. Doesn't mail.app do this?
Mail.app and most other apps since 10.8 (?) introduced badges just use those, unless it’s a more urgent thing than “you have a new message” in which case, yes, it will bounce the dock icon.
I vaguely remember it having history, but the native system does have something similar in that panel you get by swiping with two fingers from the right edge of the trackpad, in "notifications" tab.
Polite apps would just bounce their dock icon rather than stealing focus, however.