'care for' often means 'like': I don't care for chocloate much.
In British English, we can use 'sick' and 'ill' for any infection, illness, etc, but we don't use 'ill' for 'vomit'. I believe that Americans don't use 'ill' much.
'Sick leave' is when you are away from work, school, etc, because you're ill.
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