I noticed you also posted this in TAW. While such matters are usually identical between the "Eastern Orthodox" and us Oriental Orthodox, I anticipate that the answer to your question will be slightly different between the two.
According to catholic teachings, the bread turns into the flesh, bood, soul and divinity of Christ, and the same for the wine. Do ortodox think the same, or the bread is just the flesh and the wine is just the blood?
The teaching that both elements of the Eucharist contain both the Body and Blood of Christ was a speculative innovation of the Western Scholasticism in about the 13th century, after the Western church had already been divided from us (5th century) and even the Byzantines (11th century). We are not generally interested in such speculations. And we certainly resist attempts to treat it as dogma. We are content to just stick with the Apostolic teaching that the bread becomes the Body of Christ and the wine becomes the Blood of Christ.
According to catholic teachings, the bread is not more bread, it is just all Christ, and the same for the wine: it is the teaching of transubstantiation. Do ortodox teach transubstantiation, or another doctrine as consubstantiation?
We do believe that the bread becomes the Body of Christ and that the wine becomes thee Blood of Christ. If you are asking about something beyond that teaching, it is likely that we are not interested in such speculation.
Catholic tribute worship of latria to the eucharistic kinds; we kneel before them. Do ortodox tribute for it worship of latria, or better worship of dylia?
Well, one thing must be said and that is that neither we nor the Byzantines set aside the Body and Blood of Christ specifically for worship like the Latins do, as we believe that the only use established for them by Our Lord is our partaking of them. Furthermore, we do not even, as the Byzantines do, have a Liturgy of pre-Consecrated Gifts, which is the primary context in which you see them giving explicit worship to the Body and Blood. We will not shirk away from worshiping the Body and Blood of Christ in spirit during the Divine Liturgy, as we believe it is the Body and Blood of God Incarnate. But liturgically speaking, I believe there is no context in the Oriental rites for explicit/physical worship of the Holy Gifts.
What happens when consagrated kinds are left over after eucharist? Do orthodox think that they are no more Christ, or Christ is himself still in the kinds? Do orthodox preserve the bread and the wine, or do they throw them away?
We most certainly would not "throw them away". That would be absolutely irreverent. And if somehow there happened to be Eucharistic elements left after the Liturgy, I think we would be bound to believe that they are still the Body and Blood of Christ. However, as to what we actually do with them, our approach (at least from what I know of Coptic ritual standards [so I cannot speak for the other Oriental churches in certainty]) is different both from the Latins and even from the Byzantines. The Latins set aside the elements for the purpose of services for worshiping them; neither we nor the Byzantines do this. The Byzantines, however, do set aside the remaining elements in a tabernacle for later use in other Eucharistic liturgies. On weekdays during Lent they have a liturgy called "the Liturgy of the Pre-Consecrated Gifts" in which they do not consecrate bread and wine as in the Divine Liturgy, but rather use elements that have already been consecrated from another Divine Liturgy. We, however, do not save the consecrated elements in this way. When we have the Divine Liturgy, we either consume
all of the elements during that Liturgy, or the Priest takes the remnants to serve to an infirm person that day. They are
never kept beyond that day. I was confused at first when I saw a tabernacle on the Coptic altar. Coming from a Byzantine background I thought they were keeping consecrated Eucharistic elements in there. But in fact I was later informed that the only thing kept in there is the chalice.
