Genuinely curious now: If you have a cold or a cough, or maybe a slight fever, do you have to talk to a doctor to get a day or two off of work?

In my country most if not all corporate jobs offer private healthcare plans that let you have an unscheduled online chat with a doctor. You can get one day of sick leave that way. Sometimes two. If you need more you will most likely be asked to show up for examination in person.

Generally yes if you don't want your vacation quota to get reduced.

In my country we get paid 80% of our usual pay on sick days prescribed by doctor.

We also can get an appointment online and get that sick day. We can also get sick days up to 3 days prior.

Also you can get a sick day if your kid is sick.

Well, yes.

You can't just not show up for work and call in sick. Within the same day you have to call your doctor and explain the symptoms and they have to start the paperwork for the sick leave. Even if it is for a day.

For a cold or a cough you would not get sick leave, though.

It depends on the country and company, but usually there are some days you can take off without evidence, and when you need more, you need a doctor's note.

In the Netherlands: This exactly, generally after a few days you get a call by the "company doctor" (it is not from the company, but the literal translation) with an invite to discuss your health issues.

I'm in the USA and I've never had to do more than inform my employer that I'm not coming in because I'm sick. I guess it varies by employer.