If you have no problem allowing the children to be with her most of the time and she can afford to stay at home, what is the problem? Regardless if she works, as long as she has the right to receive child support you have an obligation to pay child support to her. If you believe she is committing welfare fraud, that is entirely another criminal matter. As far as whether it is fair that she doesn't have to work.... life isn't fair. You need to make decisions based on what is best for each of your children. I can assure you that child support is never enough to raise children so I am guessing she thinks what you are paying isn't fair at all.
Under Texas Law, you have an obligation to provide support for your children. I cannot tell from your question whether there is a child support order currently in place, or if you are just providing support according to some agreement that you have with your wife.
As far as amount of support, there are statutory guidelines as to how much support you have to pay, which would equal 30% of your net income based on 3 children, but the Court isn't obligated to follow that guideline and he can deviate up or down if he has a good enough reason. (actual expenses, other sources of income, cost of living, etc.)
However, even this amount of child support is most likely not enough to live well off of without working, but she may be able to scrape by on this amount while staying at home. Since she has custody, then she has to make decisions on what is in their best interest, even if that decision is to be a stay at home mom.
So, to answer your questions; no, the Judge cannot force her to work, and if you do not think that her living situation is in the children's best interest, then you should look at hiring an attorney and fighting for custody.
No, the Court will not order the Mother to get a job. If you feel that the children are not being provided for in their current situation, you can seek to modify the custody order and have them live with you. But, an order requiring someone to work is unconstitutional.
Unfortunately the court cannot require your ex to get a job. Child support in Texas is not based on the custodial parent's income but strictly on your income. This means that the custodial parent could be a millionaire or a housewife, either way, absent an agreement your obligation to pay a certain percentage is mandated by statute.
No, you cannot require the mother to work. Consider how much child care will cost for three children if she chooses to work. The family code states states the percentage that you will pay based on the number of children you have with the mother and will also take into consideration if you have a duty to support any other children. The court could take into consideration whether to adjust this amount based on needs; however, most often the court will set child support based on the guidelines. Only your income is taken into consideration in calculating the child support amount according to the guidelines, not the mother's.
This amount of child support is not much money for a person with a child or children to live off of considering rent (or mortgage), food, utilities, car and car expenses, clothing bills, school expenses, medical expenses, taxes, hair cuts, and so on.