Purpose of Dome on Tweeter...

What is the purpose of the "Dome" on a dome tweeter? Will a dented or dimpled dome effect the sound dispersion coming from the tweeter? If not, why aren't they just pushed in at the factory? Better yet, if a dented dome doesn't effect the sound, it would seem as if it's a non important part in a tweeter so why not just build a tweeter without the dome at all?

Would there be an advantage to domed woofers?
"Would there be an advantage to domed woofers"?

If it gave the behavior the designer wanted, then yes.

My guess is that there would be a tradeoff, where the dome diaphragm is probably stiffer and remains pistonic to a higher frequency, but then has nastier behavior when breakup finally does set in, compared to a more traditional curvilinear cone. Few choices in loudspeaker design are without tradeoffs.

Morel and Dynaudio make woofers whose voice coils are quite large in diameter relative to their cone diameter, so they end up being quasi-dome woofers. Both companies also specialize in tweeters that can be crossed over unusually low, and this may be an adaptation to the characteristics of quasi-dome woofers made from paper or polymers.

I think Accuton uses a concave dome profile for their ceramic-diaphragm woofers, and this would be consistent with their philosophy of maintaining pistonic behavior as high up as possible. I think Cabasse used to make shallow-dome-profile honeycomb-sandwich woofer cones, but I don't know what material they used.

I do believe EPI had the first inverted dome tweeter that was not attached to a horn
As Duke mentioned above, Dynaudio & Morel mid/bass drivers with 3 inch voice coils basically have a 3 inch dome type dust cap. I myself am a big fan of the 6 inch Morel doped paper cone. Great controlled bass from the 3 inch VC and that raised dome acts like a 3 inch dome midrange. Check out the ATC monitors, they also have raised dome dust caps.

Fritz, builder of FritzSpeakers
Speaking of domed woofers, has anyone else noticed how much the whizzer cone vibrates and buzzes? That's obviously due to higher frequencies that get in through the crossover filter. But has anyone tried to damp the woofer whizzer cone and prevent this rude distortion? I have. I used one of those little triple stacked discs Acoustic Disc thingies from Golden Sound. Surprisingly efficacious.