I think that strictly speaking, your American friend is correct in saying that the verbs “seeing” and “watching” are not interchangeable in the two sentences you presented. The sense of “I am seeing a movie tonight” is that of going out of one’s way to watch that movie somewhere; one usually has to go to a theater to watch it—as a passive spectator among several others. In contrast, “I am watching a movie tonight” conveys the sense of viewing it on one’s own accord and without going out of one’s way to do so; in this case, viewing it is more likely a solitary act in the privacy of one’s home, perhaps on broadcast television, DVD, or YouTube.
We can see that a major difference in the sense of “seeing” and “watching” a movie is the degree of one’s purposiveness and control in doing so. One “sees” a movie as a spectator with little or no control at all of how that movie is shown. On the other hand, and more so in our digital age, one “watches” a movie by looking on at it any which way—straight from start to finish, running it backwards or fast forward, replaying it as many times as desired, etc. In short, “watching” a movie is likely to be more participatory than just “seeing” that movie.
Idiomatically, though, these semantic distinctions between “seeing a movie” and “watching a movie” are often lost or not observed in actual usage. For instance, there’s hardly any discernible difference today between saying “The elderly couple sees a movie every night” and “The elderly couple watches a movie every night,” particularly if we don’t really know where they do the watching. And even if we knew better, there’s really no point now in splitting hairs when someone asks “Did you see that movie on TV last night?” instead of “Did you watch that movie on TV last night?” In these days when cinematic entertainment or spectator sports are now shown through an ever growing mix of digital visual media, it’s no longer worth quibbling over the semantic difference between “seeing” and “watching.” In this larger context, your American friend is probably too grammatically purist and passé in insisting that “seeing” and “watching” are not interchangeable at all.
As to your last question, Sky, it’s indeed grammatically incorrect to use the verb “look” in this question: “Could you please look at my bag because I am going to the toilet?”
To “look at” means to just direct one’s eye or attention to something; it doesn’t convey the vigilance needed to ensure that that thing isn’t stolen. Your American friend is right that the correct word is “watch,” which means “to take care of”: “Could you please watch my bag because I am going to the toilet?”
However, there’s a verb phrase using “look” that also means “to take care of.” That verb phrase is “look after,” and it works just as well as “watch”: “Could you please look after my bag because I am going to the toilet?”
