How was Thawn in that alternate universe with Flash? If it was only The Flash who changed the past wouldn't Professor Zoom be different as well?
* Spoilers *
To understand, you'd need to read the comic - the movie leaves quite a few things out. In the comic, Professor Zoom understood the very point you bring up... that if he went back in time and erased The Flash from existence, it would prevent his own existence as well. To get around that, he came up with something diabolical:
Zoom went back in time and killed Barry's mom (he was the unknown assailant), then hatched an elaborate scheme that essentially provoked Barry into going back and saving her. The death of his mother is what drove Barry into criminal forensics, which is what led to him later becoming The Flash. By preventing the event, Barry became a paradox - like a bubble in time - and Zoom could then kill him without affecting his own timeline. Since both men were time travelers from the original timeline, they were both equally resistant to the changes.
If you liked the animated version, I highly recommend the original comic event (it's called FlashPoint) - there are numerous differences, and the comic version has a much richer story regarding the events of the alternate timeline.
Thawn still existed because he was from the Flash's personal future. Since he has travelled back in time to be in the "Flashpoint paradox" events, he is from a different timeline altogether. That establishes the fact that he can time travel as well. So two time travellers remain the same despite the changes in the timeline. (This explanation would've made perfect sense in the Whoniverse though!). See if this answer makes sense to you.