Well, the Byzantine Rite provides one or more saints' commemorations for each day of the year; each Church has its own version of this calendar, with ranks assigned to each saint depending on his or her perceived "notability". Here is the calendar for the Byzantine (Ruthenian) Archeparchy of Pittsburgh: Calendar of Saints
The church book called the Typikon gives the rules for combining these commemorations with Sundays, feast days, etc. For example, on an ordinary Sunday, you will not normally hear about the saint of the day at the Divine Liturgy, unless the saint's day is one of a higher rank; but you WILL sing hymns in honor of the saint on Saturday evening.
Also - a minor saint who is normally eclipsed by a more important saint on the same day, may still be celebrated in a major way in churches dedicated to him or her.
Here is one source you can use: the Byzantine Catholic Church has a set of books, one per month, but the a short biography of each day's saint(s), along with troparia and kontakia. Go to Metropolitan Cantor Institute publications
and scroll down to "The Byzantine Monthly Menaion". Here is the volume for June: June menaion
Not every saint has has or her own troparion, and even when they do it may not be available in English. So the Byzantine Rite provides a set of common hymns for each CLASS of saint: apostles, martyrs, monastics, etc. These hymns can be found in the Divine Liturgies book for the Byzantine Catholic Church: Common of Saints for the Divine Liturgy
(Other books contain the common stichera for Vespers.) I am fairly sure that the Ukrainian Greek Catholic Church has similar materials in English, but I don't have them at hand.
1. There are lots of saints, and saints' hymns, in the Byzantine Rite.
2. Which ones are commemorated depend on Church, local custom, and presider's preference. Layfolk, of course, can commemorate any saint they choose!
Yours in Christ,