Each push-down operation pushes down only the changes that are made to the parent content type at that time.
To create or manage a site content type on a site, you must have Web Designer access rights to that site.
I don't have a whole lot of experience with Info Path, but then again, this is not the first Info Path form I created - I've created a couple of other forms and form libraries and I never ran into the sort of trouble I am experiencing with this particular form when I try to publish it.
It appears that some of the information is being published to the library - for instance, when I look at the library's settings, the data fields have generated the columns.
Share Point Foundation does not add the column back into the child content type.
If you try to perform a push-down operation on a child content type that is marked as read-only, the push-down operation fails unless you set the parent content type to be read/write as part of the push-down operation.
If you do not have the appropriate access rights to a child site, push-down operations to content types that are contained in that child site fail.