I think most things were explained fairly well.
Regarding what everyone was injected with...
If I recall right, Maya Hansen calls it Extremis at the start of that scene where she's explaining it to Tony.
She also goes into more depth about how it actually works... (by rewriting the human "repair" centre of the brain to make upgrades.)
As for the people, they aren't dead. They were just injured in various attacks or wars. So when they got injected with Extremis they regenerated. Though as a side effect of the version of extremis they were injected with being unfinished and unstable... not everyone can survive the process. Some of them explode violently.
Because they didn't want their work compromised, they disguised these (highly illegal) human experiments as terrorist bombings.
So that's when they created/revealed the character of The Mandarin (who was "leading" their group, the 10 rings). He was a public figure designed to take credit for the "bombings" (which were just failed Extremis experiments). This allowed the real villain to continue to experiment with Extremis without having to worry about the authorities chasing him down.
Their purpose for creating Extremis was to cure Dr. Aldritch's disabilities, which they did, but the Extremis virus was unstable and he wanted to fix that (which they tried to do by injecting Pepper Potts with it so Tony would fix Extremis for them. But that didn't exactly work...)
Once Extremis was fixed, his plan was to kill the President (in a "terror attack" by the Mandarin) and then use the completed Extremis as a bargaining chip to control the Vice President (because his daughter was missing a leg and he wanted Extremis for her) who would become president after the death of the current president.