You mention "hours" and "incense offerings" separately. Is there an Ethiopian tradition that maintains an offering of incense more like what would be seen in the Jewish temple? I know that many old Jewish customs, laws and worship have been preserved in your church due to your ancient interactions with Israel, which flowed well into the acceptance of the Gospel for the Ethiopian people.
Also, I haven't heard much about Ethiopian prayer beads, or Oriental Orthodox prayer ropes/beads in general. How are these beads constructed (lengths, materials, patterns, etc.) and what prayers are used? I'm interested to know how they differ from the EO use of the chotki and the RC use of the rosary.
Lastly, I just have to comment that I have a great appreciation for Ethiopian iconography!
The Hours (Se'at) in Tewahedo tradition are not the same as the horologium in the Latin tradition and the Agpeya in the Copts, rather its an indigenous version of the similar prayers. Unlike in the Copts, the Hours in Ethiopia is almost strictly monastic, though many in the laity enjoy it as a colloquial prayer book, and the services at the appointed times are made public.
The Incense offering is its own distinct prayer in the morning and evening in the Tewahedo tradition, separate from the Hours prayers or from the Divine Liturgy, and is offered each and every day by the clergy and by the ordained monastics within the churches. These prayers are also quite public, it is the Morning Incense offering which many Ethiopians attend each morning as their morning prayer in the Hours style. Those exceptionally pious will pray the Se'at privately before the Morning Incense. On Saint's days, commemorative Divine Liturgy services are offered after the Morning Incense (accept on fasting days, when Liturgy is offered strictly after 12pm). On Sunday services, the Morning Offering is prayed, then any scheduled Baptisms, then the "Prayer of the Covenant" (every morning, afternoon at weekday Divine Liturgies, and every evening along with Evening Incense), then Divine Liturgy.
We have a solid Judaic aspect to our worship, including the popular and theological embracing of Deuteronomic ritualistic purity in connection with the Church building, the Ark of the Covenant as an Altar Stone (the Tabot), and a general feeling of entering into Temple worship at the Church building, alongside other Judaic cultural practices inherent from Ethiopia's indigenous Semitic culture. Of course, I understand many of these Judaic aspects to also be quite common in other Orthodox jurisdictions, even in the Roman Catholics.
You may want to refer to this thread for the prayers for the Ethiopian prayer beads, or Mequteria:
These links were very accurate to my understanding of Tewahedo prayer beads. A common prayer on the beads is the "Be'ente Mariam Mehadene Kristos" (For the Sake of Mary, Christ Have Mercy on Us") and also some people rhythmically clack the beads during Divine Liturgy, but I am not sure what this signifies or its origin, I just see and hear it at services from time to time.
Do Europeans, Asians, Americans, and Australians have their own "essential" instruments?
I'm not quite sure if you are being facetious but I will take you seriously and say, what other Orthodox Christian traditions use hand drums? In all seriousness, I don't know of any others. The only indigenous hand drums I know if Orthodox regions are the Armenians who have lovely hand drums, but I am not sure if they use them liturgically or in prayer. In Tewahedo, the hand drum is the essential instrument to liturgical worship, it is crucially crucial, and I also know that the other Orthodox Christians in Jerusalem complain that the Ethiopians make a racket during our celebrations and processions with our drums. Again, do any other Orthodox traditions incorporate the hand drum into liturgical worship?
While I have enjoyed discussing the spiritual and ascetic practices of Ethiopian Orthodox Tewahedo Christianity, I think an idea this OP was trying to express were perhaps ideas like Energies and Essence theology of EO which have different explanations to the economia and activity of the Godhead in the earth, which have fundamental implications in prayer theology in the concepts of theosis and kenosis and the Divine Mysteries. OO theology is far less mechanically oriented as the EO, and we rely on Cyrilian language to guide our understandings, which are a bit less precise and have fundamentally different views on spiritual matters.