At the moment I pay for a Premium Abo and one child is using it. How can I transform this into a Family Abo and invite other members of the family? How can I keep the songs and playlists of the child who is already using it? Is it recommended to transform it just one or two days before I have to pay for the next month?
You upgrade the account of your child from Premium to Premium Family and then invite your account to the plan. The remaining Premium time will then be converted into Premium Family time, this can then bring your payment date forward.
You upgrade your own account to Premium Family and invite your child to it. When they join, their Premium subscription will be paused and will resume if they ever leave the plan (or get removed from it).
When choosing option 2, you can ideally upgrade a day or two before their Premium renews, so that only a very little amount of time will be paused. They can also cancel the subscription (preventing a renewal) before joining the Premium Family plan.