Q] Where are the Quintessons?
A] Whilst they were major villains in the third season of Transformers, the Quintessons are not in The Headmasters at all and are not even mentioned by name. There are two possible reasons for this. The first is that the writers simply didn't want to use them, and just wanted the Destrons to be the major bad guys again. The other reason is allegedly that due to an argument between Quintesson designer Floro Dery and Hasbro/Takara, the Quintesson's couldn't be used any more.
Q] Why can the Cybertrons fly now?
A] They just can. Seasons 1-3 of Transformers mentioned that Cybertrons could not fly, but the writers were never totally consistent with this. The writers for this series decided to just permanently change it so the Cybertrons could fly.
Q] Why do characters who died in the movie appear in this series?
A] The movie was released in Japan in 1989, meaning that the writers didn't know who was dead and who wasn't, but I think there's more to it than that. For a start, you'd think someone would have shown the writers the movie at some point before work on Headmasters began - after all, the Japanese intro for season 3 (Transformers 2010 in Japan) contained footage from the movie. Also, the Japanese writers had to have some knowledge about events in the movie, since Starscream, Skywarp, Thundercracker, etc. don't appear in Headmasters.
Q] Was The Rebirth ever released in Japan?
A Japanese dubbed version of The Rebirth was released on Laserdisc and later DVD. Some of the Japanese voice actors returned for this, but some characters had new voice actors. The Japanese version of The Rebirth also uses American names for some characters, such as Kup instead of Chear.
Q] Was the anime an adaption of the manga?
A] There was a Headmasters manga, but it has completely different stories. Please check the TF Wiki's section about it for more information.