I'm rather confused here... I told Yes Man that I'd help protect the president and all of a sudden I completed a quest "Don't Tread on the Bear" which means I won't be able to help them anymore. So basically I don't help both Yes Man and the NCR and get the achivements??

Yeah, in the end-game you have to side with one faction, be it NCR, Yes Man, House or the Legion. You'll have to do multiple playthroughs (or revert to an earlier save) if you want all the achievements.

Protecting the president (You'll know it when it happens) is a quest that both Yes Man and NCR have in their lineup. So you will not alienate the NCR by doing that quest because they want you to do it as well.

Failing "Don't tread on the bear" won't stop the NCR not attacking you when saving the president or in the final battle; it just means that you can no longer follow the NCR endgame path.