Federal law: whoever's got custody 51% of the time gets to claim the child as an exemption, unless there's an agreement to shift the exemption to the other parent, and the parent who's giving up the exemption for that year signs an IRS form doing so.
If any of the children are under 18, she can. Child support is determined not just on the amount of time spent with each parent, but on a complicated formula which also factors in the income of both parents, so that in some cases, a higher-earning parent may be required to pay child support to the lower-earning parent, although the higher earner also has the child(ren) with him more than 50% of the time.
To figure out your best options, you should review all the facts of your situation...
How is she "restricting" your communication and travel? How is she restricting your relatives from visiting? If you're an adult, you can go where, and when, you please, and if you have the right to occupy a home (whether you are4 a renter or owner) you have the right to invite anyone you want there.
This would be, most appropriately, a "motion to set aside" under CCP 473, due to "inadvertence mistake or excusable neglect". You should prepare the documents and get the motion filed and served as soon as possible; your time limit is a "reasonable time", which cannot be GREATER than six months, but should be as soon as you can get this done.
Notwithstanding the earlier answer (from an attorney outside of California), if you get a "quickie offshore divorce" from a place where you don't really live, and someone decides to make a fuss later, the "quickie offshore divorce" is probably NOT valid, under the California version of the Uniform Divorce Recognition Act.
You can only sue her for libel successfully if what she said was both
fact (not opinion)
Unless both of these are the case, you've got no lawsuit.
Then you have to prove some amount of damages to reputation. You say your husband "was incarcerated" so proving that he was financially damaged by what she said on a website is going to be hard, if not impossible.
Assuming you hire an attorney, sue her, and win, does she have any non-exempt assets (like multiple...
There is only one kind of trial court judge in California: the Superior Court. If matters need to be consolidated, they can be, and if third parties need to be "joined" to a dissolution case (divorce) they can be, but this is procedurally complicated enough so that you need an EXPERIENCED family law attorney to assist you.