If she has no annual gross income, can she can still claim them as exemptions and receive an income tax refund?=======>>>>>>>>>>then, No. she would have to show that she paid for the children's support in order to claim them as a dependant. Only one parent can claim the children as dependants, the parent who supports them.unless she gives you form 8332, you can niot clam them.
If she files with her new husband (and she still has no income) can they claim our children on their tax return and receive an income tax refund?=======.>>>>>>>>>>>>>yes aslongas they support them.
If the answer to the above two questions is no, do I have any legal standing to demand that I claim all 3 of our children?=======>>>>>no unless the custodial parent uses Form 8332 to do to release a claim to exemption for the children so that you, the noncustodial parent, can claim an exemption for the children.