If this play would have been part of a trilogy, are there any theories as to which part and what the other parts would have covered?
Kitto's Greek Tragedy is a really good starting point for this.
Essentially, although tragedies were presented in groups of three, along with a satyr play, at the festival of Dionysius in Athens, there was no necessary thematic connection among the tragedies. Aeschylus was distinctive in offering connected dramas -- the Oresteia in particular is exquisitely crafted as a unified trilogy, but it was not Sophocles' practice. The Theban trilogy is a misnomer; the three plays by Sophocles on Theban themes (Antigone, Oedipus Tyrannos, Oedipus at Colonus) were presented at different times.
Since the overwhelming majority of Greek tragedies address the Trojan war or the returns (or events immediately afterwards), with the exception of the unique experiment of Aeschylus' Persians, there will be subject matter links through a shared body of source material. For example, one could say Philoctetes and The Trojan Women address events related to those in Ajax, but there is no evidence that they were a trilogy. Unfortunately, the date of the original performance of Ajax is unknown, as are the other plays performed with it.