> Is sick leave paid in US by law (till certain extent of course) from some employer's mandatory insurance, or some sort of corporate perk like extra paid holidays? And what about when having sick child?

For the kinds of jobs we’re talking about here (corporate Amazon, college education required, salaried) you basically just call in sick and that’s it. You didn’t work but you still get paid your normal salary. This is the flipside of salaried work obviously — just as there may be times you work over 40 hours and don’t get additional pay, there are times when you work less than 40 hours and still get your regular pay.

Now all of this depends on circumstances like your manager, but in general, in these kinds of jobs, no one is running around tracking your sick days. If you’ve got something more substantial going on then you’ll have to take short-term or long-term disability however.

That's not how Amazon operates. You have paid personal time that you must use as sick time. You get 6 days per year or whatever the legal minimum is in your state if it's over 6

The 6 day thing may be corporate policies but there are (were?) absolutely teams where you just call in and nobody really cares what the personal time balance is.