If you're talking about a specific group of children, then the article is necessary. In your example of a caretaker, assuming you're giving instructions on how to care for your children, then yes the article is necessary.
In the unlikely event you're lecturing caretakers on how to care for children in general (perhaps you're some kind of trainer or instructor giving career advice for example), then you could leave off the article.
As to your second question, either one is acceptable usage nowadays. In the past, only 'his or her' would have been deemed acceptable on grounds of number agreement, but the plural possessive is widely used and accepted now. I doubt you'd find many who'd consider it grammatically incorrect these days.
Edit: Perhaps I should amend that last statement to 'many who would object to its usage'. There probably are still a fair number who'd consider it grammatically incorrect, but acknowldege the current trending change in usage.
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