Anyone who uses picking on a new nurse in front of other staff members as a way to boost his/her ego is an insecure jerk, and it sounds like that's what he was doing. Really, why would anyone go out of their way to make someone feel stupid or inadequate unless they had a much deeper problem with feeling inadequate themself?
Try to put this incident behind you and remind yourself as often as you need to that you are a competent nurse and every nurse in the world had the exact same learning curve when they were new. The NCLEX is HARD, don't kid yourself. If you were stupid you would not be a nurse at all right now. Believe in yourself and your ability to make the world a better place with your skills and talents and try as hard as you can to disregard jerks like this who want to make other people feel as badly about themselves as they do. As far as the nursing assistant smirking, he/she is probably just jealous because you are a nurse and he/she isn't. Jealousy can make people act really stupid sometimes.
I have been an RN for 22 years, and I still feel stupid taping report! I was so glad when our facility went to bedside reporting and I could stop using that stupid, intimidating tape recorder, but since you have to for the time being, try to look at it like this: when you listen to report, whether it's from a new nurse or an old nurse who's been around since the Reagan administration, you're really just listening to the information that they're giving, not the way that they're giving it, right? The day shift is probably not making fun of your report at all, just gathering the info like you do. Try not to let them intimidate you, because every one of them was once a new nurse who felt the exact same way that you do.
Hang in there, it will get better and don't beat yourself up! You sound like a caring person who only wants to do a good job, and you will be fine, I know it.