This whole thing with finding parents/siblings.... I'm very new to it.... Can anyone please explain how it works, and what is the purpose?

The purpose is ostensibly to make sure that you really want to be a family with the other person, with the true reason being that they don't want you to adopt 50 kids in a day and become an instant Great House.

Anyways, two parents can CONSIDER a child, and after that period is done they can fully welcome the child into their family (at which point the child gets their family name, and so on). It's also possible to form a BLOODBOND with another (also taking 2 IC years), which allows you to create a family name. The two siblings can't consider, however. Each one would need to marry and then consider with their spouse, which also takes time. (Both for the marriage itself, and the considering)

And it's not adpotion, you really are their child if they consider you and the consideration goes through