Parents are detained. What about underage children?

The public security organ that make that detention decision has the obligation to look after these children,

Failure to perform this obligation constitutes abandonment.

This is a real case.

The end result was tragic. The mother was detained and the child starved to death. This case caused a sensation at that time, and all jurists agreed that it constituted the crime of abandonment and death with a reprieve.

Now that a person is to be detained, does the police have to show any proof to detain a person just by verbally saying the other person's situation?

Just have a detention permit.

Whether you can ask for bail after 24 hours.

You can get bail at any time, but whether you agree or not is a matter for the police.