To hit the last post:
Exaposilaria are a group of liturgical hymns sung at Orthros in the Byzantine tradition. They are usually in one or two musical "keys" out of they eight used in the Byzantine liturgy. Most of these hymns are thought to have composed by monks either at Saint Sabbas monastery near Jerusalem in decades following the Islamic conquest, or by the monks of St. John Studios in Constantinople the decades of/immediately after Iconoclasm.
Re: the OP:
Some exapostilaria are dictated by the Octoechos, some by the the Menaion/Pentecostarion/Triodion, or on "regular" weekdays the Horologion. I believe all the Sunday Resurrection Exapostilaria are to the 2nd mode tune "Τοῖς Μαθηταῖς συνέλθωμεν", the "regular" weekday exapostilaria are either 3rd mode "Ὁ οὐρανὸν τοῖς ἄστροις" or 2nd mode "Γυναῖκες, ἀκουτίσθητε". Most of the hymns of the Octoechos are to either of those melodies, but not all (e.g. Ascension).
As to the why, you'd have to ask the monks, and you'd have to wait after the Second Coming.
My own opinion is that, in their current form*, these melodies were made to have a special acclamatory character appropriate to that part of Orthros, and that subsequent hymnographers (principally in the Menaion or in the numerous additional acclamations outside the mention) are simply following the original pattern.
* That the current melodies reflects the original melodies cannot be taken for granted. Capella Romana did a CD of some of the melodies used by my own "ethnic" jurisdiction, the Italo-Greek monks, written down probably within 2-300 years after the composition, and some of the melodies are indeed quite different.
As another aside, it's kind of hard to appreciate Orthros in the parish because the current practice is a monastic service, is meant to be 2.5-3 hours long, builds on everyday familiarity, and at least in current practice is mostly recited with the occasional sung hymn. The exapostilaria are especially acclamatory because they end an hour or so long, mostly recited group of canons.