Under the law, you have to notify your child's mother if your current wife is on the sexual predator list. You also have to let her know where you live, what your phone number is, where you work, and what that phone number is. You do not have to be friends of facebook, notify her about your schedule when you son is with you, or give her any other information about your current wife. She does not have the right to limit your visitation on her own, only through a Court Order.
That being said, you should probably tell her what you would want to know. If you want to know when the child goes out of town with her, then you should tell her when you go out of town with the child. If you want information on her new significant other (either now or in the future) then you should give her some on yours. Be reasonable. What elementary school was attended is ridiculous. Just try to think of what information you would reasonably want.
Ms. Brochstein was correct when she said that you will have to co-parent with this ex until the child is 18, and maybe beyond.
Don't let her screw around with your visitation. If she denies visitation, take her to Court. Talk with a good family law attorney about your options. Remember that if she denies you and you do nothing, the Court will assume you do not care.
You need to hire an experienced family law attorney.
You need to have the court ordered visits with your son.
She can go and verify that your wife does not have a criminal record - it's public information. Of course, she will have to go down and pay for it herself.
You do not have to be her "friend" on Facebook.
I urge you to learn to co-parent with the mother. The two of you are going to be parenting together for the rest of your lives. You will be co-grandparenting your grandchildren.
Perhaps going to a mediator and talking might be helpful. Or, counseling for the entire family might be appropriate. The adults need to learn how to get along - it's in the best interests of the child for the parents to get along.
She can do an independent background check or send "discovery" asking certain questions about your wife. Questions about her childhood are hardly relevant unless she has a juvenile record. Situations like this are fairly common unfortunately. Sounds like the CP has control issues and doesn't like another woman taking care of her kids. Judges see this type of behavior all the time. It's up to your attorney to object to the relevancy. As far as FB, block her. A lot of people have trouble when the other parent moves on. Good luck!