I want to know if it is possible to adopt a child without installing Hearthfire, and how do you adopt the kids that aren't at Honorhall?
If you don't have Hearthfire, then you'd have to install mods like Adopt a child to allow adoption of children. This mod, in particular, will allow you to adopt a child from the Honorhall Orphanage in Riften.
I've checked out Skyrim Nexus and it seems that the current available mods there that allow adopting of kids not from Honorhall require that the Hearthfire DLC be installed.
It's possible to adopt children without Hearthfire [WARNING: PC ONLY! IF YOU'RE ON XBOX, YOU CAN ONLY USE HEARTHFIRE FOR THIS METHOD!]. There is a mod called Adopt a child mod which allows you to adopt from the Riften Orphanage, and same requirements. There are also some adoption mods like Clover the Orphan which allows you to adopt Clover.