Yes, born-again Christians still make mistakes. One is not excepted from the propensity to make mistakes merely because one subscribes to a particular religion, philosophy, or world view. Every human being has only a finite number of experiences and a finite amount of intelligent. To say that any such human would be incapable of mistakes is asinine.
Yes, born again people still make mistakes, because they are still sinners. Even Peter denied Christ three times. Everyone is a sinner and will always be a sinner except for Jesus Christ, because he was perfect. We can try to be better people and follow God's commandments, but we are all still broken people, and we will never stop sinning.
A born again Christian should not be saying that they do not make mistakes or commit sins. They are trying to say that of course they do because everyone does. However, they say they are not worried about that because they have been saved by Jesus Christ and his sacrifice for them.
Yes, I think people who are born again still make mistakes. Everyone makes mistakes, even born again people. The important thing is to grow and learn from your mistakes and try not to continue to make the same mistakes over and over and over and over and over and over again.
Born again people aren't perfect and they tend to make mistakes like the rest of us. A born again Christian doesn't automatically become some omnipotent being. Without a doubt, they're just as likely to make mistakes as someone who isn't religious at all. Mistakes are a part of being human.