A good example of how language isn’t just about the grammar — the social context matters. I do this as a native speaker of American English, and the idea behind why is that I (we) want to sort of soften the effect of the possible suggestion. We want to leave open the possibility of some other option for our conversation partner, even if we probably know the answer actually is “yes” to the main thing we are asking about. We don’t want to completely impose our guess/hunch on our conversation partner. (This is a cultural value across the English speaking countries.) In the context of the first example you give, maybe one friend, Brian, is having a dinner party. And maybe he knows that Kevin, the friend he is asking to come to the party, has separated from his wife, Sarah. So it’s awkward for Brian to just assume what is happening with his friend’s life, and definitely to communicate any assumption to Kevin. Give Kevin the option of a chance to say, “Sarah has something she’s doing that evening, actually.” Let him save a little face.

In general, if you hear native speakers using their language in a certain way, you should feel free to speak in the same way. You are right: they are using incomplete sentences. It isn't formal, "correct" English. You might not want to speak that way in more formal situations. However, it seems to be OK with the people who are speaking that way.

(moved comment to answer)

It is perfectly acceptable to use those phrases so long as you add the question at the end. Are you coming with your wife or not? Are you here by yourself or not? You want to grab some coffee or not? Are you coming with your wife or with someone else? Are you here by yourself or with another person? You want to grab some coffee or something? Are you coming with your wife or you children/kids? Are you here by yourself or you lover/girlfriend.? You want to grab some coffee or a meal? Are you coming with your wife or alone? Are you here by yourself or with a companion/company? You want to grab some coffee or a bite to eat? ETC there are countless things you can add on for individual situations.