I really do not know much about the eastern rites within the Church.
What sort of Eucharistic traditions/practices are there within these rites? Specifically, I am wondering if they practice eucharistic adoration, and if Mass (or Liturgy - I forget the proper term in the eastern rites) is celebrated daily.
I will speak to the Byzantine tradition, as far as a can. Others may wish to correct me, or enlarge on the subject, and that’s fine.
Divine Liturgy is only really possible if there is a congregation [someone to make the responses], the tradition does not allow for private Mass. This is one of the reasons Divine Liturgy is not offered daily everywhere. Secondly, the altar must be fasting [one liturgy per altar per day] and the priest must be fasting including he abstain from conjugal relations. Therefore since most priests are married, unless there are several married priests available to a parish or a priest-monk assigned, it is not likely that there will be a daily liturgy scheduled in a parish. The parish also has to be big enough to have some kind of congregation, at least a cantor of some sort, available every day.
These limitations do not apply to a monastery, when the priests available are always celibate and the congregation is always on site. So there can be daily divine liturgy, and in many places there is, but not in most parishes. I am not aware of any ironclad rule in the tradition for specific days during the week other than each Sunday and all Feast days, although I am aware there is a prescribed schedule for the
Pre-Sanctified liturgy served during Lent and Holy week before Pascha.
On the question of Eucharistic Adoration, the concept is different.
It is thought of differently. The eucharist is in the eating [the act of receiving, chewing, swallowing]. Put another way, in the Byzantine East (and perhaps elsewhere) the eucharist is a verb, in the Latin west it tends to be thought of as a noun. In truth both senses are correct, but the emphasis is different. So certainly there is intense adoration
while receiving, but otherwise not so much. There is no special ceremonial devotion to the reserved species.
This is one reason the bishops meeting at Brest, who agreed to come into communion with the Pope, made a point of not being required to participate in Corpus Christi processions [the fact that they made a special mention of this suggests that they were being pressured on this point at the time].