Can you adopt children in Skyrim?
With the official DLC Hearthfire it becomes possible to adopt children.
You need to have the Skyrim add-on "Hearthfire" installed. It's on Steam for around $5, if I'm not wrong. Once you have it installed, random new children will appear in towns without parents and you will have dialogue to adopt them. Also, a courier will give you a letter saying the Honorhall Orphanage in Riften is open for adoption(provided you have previously killed Grelod the Kind).
Replying as Dragonborn, Harbinger or Arch-Mage will result in amazement and disbelief from Constance. so she will allow you to adopt one of the children from the orphanage. just talk to one of the children and they will have a dialogue option to be adopted (also make sure you have a house with a childrens bedroom which can be bought from a jarls steward) hope this helps
No it is impossible to have children on Skyrim... the main reason for this is that the Bethesda may have felt that there is no real reason to have children in a game like Skyrim and they may have felt letting the Dovakin have a family would slightly be copying Fable (because the Hero in Fable can have a family)