The main issue with the list of genres in the question is that it mixes terms that have been used since antiquity (especially tragedy and comedy) with terms that are particular to specific periods or authors (history plays, epic theatre, theatre of the absurd) or to specific cultures (mostly Western, with the exception of classical Sanskrit theatre).
In Europe, the terms "tragedy" and "comedy" have been around since classical antiquity. Aristotle's Poetics distinguished between two types of drama (Poetics, 1448a):
It is just in this respect that tragedy differs from comedy. The latter sets out to represent people as worse than they are to-day, the former as better.
However, the continued use of the term "comedy" since antiquity should not mislead us into thinking that the term has always had the same meaning. Many modern readers have difficulty seeing some of Shakespeare's plays as comedy where Elizabethans had no such problem; see for example, Troilus and Cressida, Measure for Measure and All's Well That Ends Well. These three plays have sometimes been categorised as problem plays, but there is no accepted definition of that term, either. (For example, some scholars included the tragedy Hamlet into the problem plays, so "problem plays" does not seem to be a genre in the same way that "comedy" and "tragedy" are considered genres.)
Tragedy is a genre that has not always been around (see Why did attitudes change towards tragedy?); in fact, it seems particular to a few specific periods in the history of Europe.
J. L. Styan puts it well when he writes in The Dramatic Experience (page 109):
Today we no longer deal in tragedy, it seems, but in problem plays, propaganda plays, modern morality plays, plays of ideas, or simply 'dramas', and we are at a loss to explain our response to them as either tragedy or comedy. With the movement for naturalistic drama, dramatists imposed upon themselves the limitation of being as like life as possible, and refused themselves the heightening of tragedy and the exaggerations of comedy.
Tragicomedy doesn't have a formal definition that works across the age, i.e. no definition that goes beyond "a literary genre that blends aspects of both tragic and comic forms" (Wikipedia). (Shakespeare's tragedies contain elements of "comic relief" (a misnomer according to J. L. Styan) but are not categorised as tragicomedies for this reason.) Cuddon also notes that tragicomedy has become hard to distinguish from black comedy (a term coined by Jean Anouilh).
Epic theatre is mainly connected with Bertolt Brecht and the 1920s–1940s; other examples include Vatermord (Parricide, 1922) by Arnolt Bronnen and Fahnen by Alfons Paquet. In other words, epit theatre is a very "localised" term both in time and in the literatures in which it was used (i.e. German literature).
Opera seems to have originated from "chanted tragedy" (Cuddon: entry "opera") but is usually ignored by literary studies in the West. Even the highly regarded Mozart librettists Emanuel Schikaneder (The Magic Flute) and Lorenzo Da Ponte (The Marriage of Figaro Don Giovanni Così fan tutte) are ignored in books on the history of literature. For example, Fritz Martini's Deutsche Literaturgeschichte mentions Mozart a few times but for reasons that have nothing to do with the libretti for his operas. When opera is mentioned at all, it is typically because the author was well-known for literary works that are independent of collaborations with composers. Examples inlude Bertolt Brecht's collaborations with Kurt Weill (The Threepenny Opera, Rise and Fall of the City of Mahagonny) and W. H. Auden's collaboration with Igor Stravinsky (The Rake's Progress. An exception to this tendency is Richard Wagner; Fritz Martini's Deutsche Literaturgeschichte still points out that a discussion of Wagner's work is strictly speaking beyond the scope of what is usually covered in a history of literature (pages 396–398).
Opera also means something totally different in China, where traditional Chinese opera combines elements of music, song, dance, martial arts, acrobatics, costume and make-up art. (In addition, it has many regional genres or styles, of which Peking opera is only the most famous one.)
The main lesson from this answer should be that genre theory is complicated because authors are constantly trying to find new and effective ways to capture the theatre audience's attention and their efforts constantly subvert existing genre conventions. For this reason, each effort to define genres needs to take into account both historical conditions and the literary tradition (English literature, Italian literature, Sanskrit literature) that is being studied. A catalogue of all genres across all eras and languages would be of limited value without explanations of what the respective terms mean or meant in different literary contexts.
Based on this, it is not possible to categorise all forms of drama as one of the types in the list "tragedy", "comedy", "tragicomedy" and "melodrama".
- Cuddon, J. A.: The Penguin Dictionary of Literary Terms and Literary Theory. Third edition. Penguin 1992.
- Martini, Fritz: Deutsche Literaturgeschichte. Sixteenth edition. Stuttgart: Kröner, 1972.
- Styan, J. L.: The Dramatic Experience: A Guide to the Reading of Plays. Cambridge University Press, 1965.