Regretfully, there is no accounting rule for child support in Illinois. What's done is done, and all the more reason to have an attorney during the initial support proceedings to ensure you aren't taken to the cleaners and/or to ensure if it's a guideline order, that the percentiles/deductions are being properly accounted for.
Unfortunately, it happens that child support is used for more than support of a child. The way the law is presently written (hopefully to change soon) there is little that can be done about it. If the father is paying too much, then he should return to court to seek a reduction. The father needs a lawyer to best guide him through the process.
You MAY have an argument here. You could possibly put together a two-pronged attack: 1) the child has a right to receive support from BOTH parents (not equally, of course, but from both parents); and 2) the amount of support is so great it creates a windfall for the mother (and that can be corrected).
I'm skeptical of your fact pattern, however. I doubt she's able to live independently solely off some luxurious child support award. From what you post I imagine Lovey Howell sauntering around her mansion in a silk housecoat smoking imported cigarettes with European nannies tending to the baby. In reality, there are probably other factors; like, she's a student and has some financial aid; she's living with her parents or other family members; she's worked for a while, has some savings, and can afford to take a little time off to be with the baby; she has other wealth from which to draw.
Whatever the case, you'll be hard-pressed to prevail on the above one-two-punch. It's possible, but not easy.
As for the name; he's the father, why shouldn't his child have his last name?
Finally, you can obtain a support order for your children so that money comes out of your husband's paycheck first. That will reduce his "net income" and will reduce the child support paid to the OW. This should have been done when he was in court on the support order. Was it? You can still take action, if not.
Questions? Call: 312-987-9999 -- no charge, no obligation.
Child support should be based on your husband's net income alone, not your income combined with your husband's income.
The unfortunate part of this is that you will need to go to court to change the child support. You would have to prove your husband is paying significantly more than the statutory guideline support based on his net income (alone) and argue that she has an equal responsibility to financially support the child. Since the child is 6 years old and in school at least part of the time (I assume), there is no reason the other woman cannot get a job.
Additionally, you cannot control how the other woman spends the child support she receives, even if it does provide a windfall for her. There is nothing in the law that allows you to do so.
Nothing in this post creates an attorney-client relationship. This post is not meant to serve as legal advice.