Cool - no XSI experience here, only Softimage last century for a few days! It'll probably be best for someone else to translate all the Xsi concepts, but I'll try, though we are probably entering the area known as ‘the wall of confusion’
Make sure to check out Jordi's Xsi-Houdini guides! They are very popular.
The 3 questions seem to show a that Houdini has a different approach to Xsi, at least in this bit.
“subnet_Polygon_model_in_here” isn't a child of box- it's the opposite
box is in the directory of “subnet_Polygon_model_in_here” - this the directory style structure of Houdini, not a parent-child relationship
The issue of dragging a subnet to the Obj is based on the same thing- the obj node is the base of the file system - it's effectively the root in a unix operating, but only for objects. Shader, renders, and the compositing network, also have their own areas, but, this too can be somewhat changed with Managers
Third question, that sphere is the object node - so you are effectively trying to copy that into the geometry area - which is not possible - you would go into the object node and copy the geometry node out and then paste that.
As far as I know Houdini doesn't use a ‘GroupGeoApprox’ type operator that Xsi does. It appears it affects anything you attach to it.
The subdivision you're seeking seems to be not a subdivideSop but a render time subdivision, that is easily achievable by the ‘Render Polydons As Subdivion (Mantra)’ checkbox on the Geometry node/Render tab. The viewport wont show the subd but Mantra will render it perfectly smooth.
It appears the Xsi ‘Explorer’ equivalent isn't the Tree view in Houdini, it is much closer to the List view set to Input/Output (see screengrab - this is available at the top of the Network View) I do remember every Xsi user looking for the same thing many months back. The Tree View is like a Os system file browser - you can also use the Textport to see this- try ‘ls’ to list the scene, ‘cd’ to change directory etc.