First, the IRS will have a problem if you excessively overpay your child. For example, if I could pay the neighbor kid $20 to mow my lawn, I can't pay my own kid $500 for the same thing.
Second, it would need to be ordinary and reasonable to deduct as an income expense. If you were a photography business and wanted to use your kids' baby pictures as marketing, that would make sense. If you run an auto detailing shop and put the picture of your kid up in the customer lobby, not so much IMHO.
Third, yes, they can open a custodian Roth IRA and contribute to the amount of their earnings (up to the limit of $6K per year or whatever it happens to be).
Fourth, you're incorrect about the requirements to file a tax return. They will be required to file a tax return if they have self employment earnings of $400 or more. See the instructions for Form 1040, "Do You Have to File?", Chart C, item 3.
Fifth, they will pay both halves of self employment taxes via their Form 1040, Schedule C, and Schedule SE. They'll likely get an adjustment for 1/2 of the SE taxes on Schedule 1 line 15. There may be a partial exemption for this since it's your kid working in your business, but I forget those details.
Sixth, you will have to issue them a 1099-NEC if you pay them more than $600.
...
I had all three of my teenager kids open custodial Roth IRAs at Schwab. They contributed based on things like house sitting, pet sitting, mowing lawns, and other similar typical teenager kid jobs. The oldest is now graduated college and in the workforce and I think maxes out his contribution every year. The middle contributes about half the time based on part time work he does while in college. The youngest usually contributes all of their earnings (from part time work in on-campus jobs).
I usually do a thing at Christmas where I'll either match or gift to them based on their Roth contributions. So if they contribute $500 I'll give them $500. Of course, they still have to have earned income to make the contribution. Last year I made those gifts in kind from my taxable account so that we, as a family, could defer the capital gains which would have otherwise been due. This also means all three of my kids now have taxable accounts at Schwab.