I met little Sofie in Windhelm and there was an adoption dialogue option. But I don't have a house, and I haven't killed Grelod.

So 1) can I adopt a child that doesn't live at Honorhall before killing Grelod?

and 2) Can I adopt a child without having a home? If so, would the child follow me on my travels?

1) Yes
2) Nope