Is the dictator game/dilemma studied by game theory or by decision theory? I'm not sure because of the lack of other interactive agents.
I would say that the dictator game is of interest to a decision theorist more so than to a game theorist, although the dictator game can be formally modeled in a game theoretic framework.
Decision theory studies how behavior, especially those that deviate from the rational ideal assumed by standard economic theory, can be modeled, as well as the implications of these behavioral models.
Game theory, in contrast, focuses more on the interactive aspect of decisions. The fact that the dictator game can be formalized as a game has more to do with the broad applicability of game theory than the dictator game's intrinsic game theoretic value.
Based on the purpose of your research, you should be looking up references in the behavioral/experimental literature.
Yes, the game can be studied in game theoretic framework. The working assumption is that the agents are rational and are maximizing utility. It is certainly a game - a sequential one to be precise - solved easily by backwards induction (as with many other games to be honest).
Some extension include finitely or infinitely repeated version of this game, a dictator game with incomplete information (about the receiver's preference, etc.
Of course decision science study variation of this game too. It's a common introductory example for behavioural economics.
It is not only a question of belonging to some area. However, in a strict sense, it does not belong to Game Theory because only one of the players has strategic reasoning (the other simply accepts what is given to him), even if this is a variation of the ultimatum game.
Nevertheless, this is the case when the game is played once.
If you run an experiment where players interchange roles, like it is a common practice, then you include strategic interaction, because players recall past events and take that piece of information into consideration when taking the role of dictator.
The usefulness of the game and these kind of studies, I suppose, you know as well as I do (e.g., analysis of psychological factors as revenge, punishment, and fairness)