Yes, the W-10 is strictly for dependent care providers and W-9 is for other independent contractors.
There should not be an issue for your client as long as they use the W-10 going forward.
The quarterly installments are based on the self employment tax and the personal income tax. The self employment tax is 15% and your personal income tax is graduated depending on the income. The Form 1040-ES does have the rate schedules and a worksheet.
Your client keeps a copy of the W-10 and the IRS does not get a copy unless they ask for it. It should always be kept with your client's records.
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