Do people who commit evil acts believe that they are doing the right thing?
Sometimes. Not always.
Everyone fundamentally believes they are doing the right thing. They will tell themselves that the law is wrong, the victim deserved it, other people will benefit more than the victim suffered - but it takes a literal insane person to say "I'm evil."
Almost always, though many know and simply don't care.
Yes (for example fanatics and terrorists) and no (antisocial personalities who have no empathy and don't care about right or wrong, or compulsive psychotic killers).
That's what I was going to say.
Some do and others don't. It depends on their mental state.