Yes, it includes your spouse and any number of children up to the age of 26 I think. So the classical family model with a stay-home spouse and children will be covered by the breadwinner's health insurance.

If any family member works full-time, then they have to cover their own health insurance. Exceptions apply.

If you make more than 40.000 EUR / year you have the option to get a private insurance policy with more "benefits". Then, the employer will pay out the amount that went to the public health system and you will use it (and whatever you can afford on top) to your private health insurance.

When you go private, you're in the wild west, meaning you'll need to be extremely cautious of what contracts you sign. Or you'll end up with a 3000 EUR per month plan at old age, with no way back into the public system.

Many people end up poor after retirement because their whole 401k and savings go towards the health insurance which is way to luxurious for their needs and isn't capped.