1) The more I think of it the more I think “stop by my office” is correct, which inclines me to my second supposition, that it’s a form of “pass by my office”, where “by” is used in the sense of “near”. I’m still not sure about this.
It could also be influenced by the adverb sense, perhaps also with some influence from the German/Yiddish, so it is being used as a preposition in this one instance. Still, this usage is confined to a location, not a person. So even if one can say “by my house” with verbs other than “stop”, one could not say “by me”. That is just an anglicization of the German/Yiddish “bei mir”.
2) Again, that definition specifies that it is an adverb, not a preposition. “By” is the way in which you are stopping. Its referent is the verb, not a location. Its definition is “at or to another’s home”. So it’s not “please stop by my home”, it’s just “please stop by”, which automatically implies the location where the person is to stop by.