Would a billionaire pay a lot more in child support as a mere millionaire or does the law at some point say “This kid’s getting enough money.”?
It depends. 
On the individual judge, on the kid’s needs, on the quality of each party’s lawers, and on the parties themselves - their willinness to cooperate (or not), and so on. Plus external and complicating factors.
IOW, there’s no real way to say, from what you’ve posted.
Generally yes, the more you make the more you pay. The judge will base the award on the “lifestyle” of the family and the “lifestyle” the parents can provide him with and expect the kid to have if the child was living with the parent.
So it’s up to the judge and if Mr Billionaire thinks it’s too much he can appeal 
OTOH - I believe from what I’ve read, that certaily the court is not going to award “30% of your take-home pay” or anything near that if you are a billionaire. Usually the lawyers for the two sides come to an amicable arrangement that means the child lives a very cushy lifestyle. If there is any disagreements, they lawyers will get rich if nobody else does. You don’t want to mess with someone who has very expensive and clever lawyers on retainer - for example, I bet with a nsty divorce, they would end up with some deal where any money goes into a trust and the lawyers would have to approve anything over $X. The ex is not going to be flying on private jets on a whim. Of course, ex-wives of billionaires usually settle for some cushy supported lifestyle of their own, and the kids get a trust fund.
Depends what you mean by “millionaire”. A million dollars is not a lot nowadays, so someone with $250,000 income would be a millionaire in actual assets pretty soon, unless they were pretty poor money managers. What would you expect to pay in child support if you made $250,00 a year?
In Canada, generally alimony is defunct; a wife is given X years (usually 5 max) to get back into the workforce, unless it’s something like the 30-year stay-at-home wife who helped develop the guy’s career, business, and raise the 5 kids; then she gets an income for life. Child support below a certain amount ($200,000?) is read off a guideline table - $X per child until 18 or finished college. Beyond that level, they can usually afford the lawyers to negotiate.
of course, most jurisdictions have a 50-50 rule unless tehre’s a prenup, so a lot of ex’s have pretty much set amounts they will get too; if that did not include a “for the kids” deal, it would be a pretty incompetent lawyer.
As with all negotiations, if you don’t settle, the judge will decide and odds are at least one side will not like the settlement. As someone said about Rudi Giuliani and his divorce, when the wife asks for $30,000 a year to support the dogs (yes, she did), he’s best off to pay it and move on - it’s chump change compared to what his legal bills would be.
In some states, there are caps on child support.