Daycare is not two words and it is not 'on' word. It is one word. That's in my part of England anyway. Does it matter? You could have day-care although the provided spell checker doesn't like it presented that way. But presumably you're not going to be guided by the spell checker otherwise you wouldn't have needed to ask the question. As 'it' is not a fundamental word you could I suggest do what you like with it (them). We will all understand, providing you spell each of them this way.
Two words - "one day".
'Good Day' is two words.
Just one word.
Race day is two words.
Health care is two words, however when used a modifier (healthcare provider) it's one word or used with a hyphen.
health care is two words health is one and care is two.
Two words - "one day".
The term "childcare" (child care) may be spelled as one word or two, and slightly more rarely as the hyphenated adjective child-care.
two
"all day" is typically written as two separate words.
'Good Day' is two words.
Just one word.
"Any day" is always two words. "Anyday" is incorrect.
Race day is two words.
3.
Health care is two words, however when used a modifier (healthcare provider) it's one word or used with a hyphen.
Skin & Care are two different words but people often use it as a single word "skincare", sometimes using a - in-between "skin-care". So, you can use it according to your need.