I am wondering how many children in total I can adopt at any given time. If the number is dependent on where our home is, then which home allows me the most?
The cap is two, regardless of which property you use.
As a side note, I think you might even be able to divide the children between properties. Like having one live in Proudspire, one live in Lakeview, etc.
By default you can adopt:
However, there is a list towards the bottom of the Skyrim wiki, where it outlines different children that will actually get sent to orphanages and become adoptable if you murder their parents, which is hilarious.
By default, Hearthfire will only allow you to adopt up to two children. If you are playing on PC there are mods to allow for more. However, in order to adopt you must have a house (either a homestead or one of the houses obtained by being thane of a given city), and the house must have a bed and a chest for each child. This means either buying the Children's bedroom package for a house, or building child's beds and chests in either the main hall or the optional bedroom wing (The dresser between the two beds counts as the chest for both children in the main hall).
There are four children available for adoption by default:
As well as an additional four* from the orphanage in Riften once Grelod the Kind takes a permanent vacation:
The Elder Scrolls Wiki has a full list of adoptable children, and other information related to adopting.