I hope you don't mind me coming by without calling like this. This fourth example is commonly said. However, it is not correct. Correct: (and this is what I would say) "I hope you don't mind MY coming by without calling like this". You see, you are not saying "I hope you don't mind me".....your point is, "I hope you don't mind the fact that I'M COMING BY WITHOUT CALLING FIRST, so you say it the way I said it.."I hope you don't mind what I'm doing....my coming by without calling.

If I showed up at someone's house without calling first, I would say, "I hope you don't mind my dropping (or stopping) by unannounced (like this)." You could also substitute "my coming by," but "dropping by" (or "stopping by") emphasizes even more that the visit was unexpected or spur of the moment (=spontaneous). The proper construction is "my dropping by," because what folllows "mind" is going to be a direct object, so it should be a noun or a pronoun or a clause, such as "I hope you don't mind that I'm dropping by unannounced." Since "dropping by" is treated as a direct object (it's a gerund/noun form of the verb), it is preceded by an adjective, in this case the possessive pronoun "my." It would also be reasonable to say, "I hope you don't mind my dropping (or stopping) by without calling," but then if you were going to say "like this," the sentence order should be, "I hope you don't my dropping by like this without calling (first)." The "like this" refers to dropping by, and if you put it at the end of the sentence, it sounds like you "didn't call like this," which doesn't make sense and sounds very funny to the ear.