So the topic of the resurrection of the body was very much a hot topic of debate during the first couple centuries of the Church, especially as a point of contention between pagans and Christians. Pagans objected that the elements which compose the body become scattered after death: how could they then be gathered back together to form the body. Furthermore, if some of the matter of my body comes from somebody else who died before me, who gets that matter in the resurrection? The standard Christian response to this was to appeal to God's omnipotence, the argument being that a God that can create everything ex nihilo can surely recompose our bodies in the resurrection. For an example of these arguments you could read St. Gregory of Nyssa's On the Soul and Resurrection, though that is only one among many patristic discussion of this topic.

The nature of the resurrected body is also of great debate for early Christians. There is a tension in that Christ's resurrected body, as described in the Gospel accounts, seems much like ours, although it can walk through doors. On the other hand, in 1 Corinthians 15:50 the Apostle Paul writes: "I declare to you, brothers and sisters, that flesh and blood cannot inherit the kingdom of God, nor does the perishable inherit the imperishable."

The great thinker Origen tries to resolve this by arguing that Christ's body after the resurrection was not his 'final form', so to speak, but was rather an intermediary state between his earthly and resurrection bodies. Origen also has a concern that, since our bodies are constantly changing, when resurrected which body will I have? The one I had at age 25, 26, etc. His solution is to say that we will have the same eidos, which could be translated as something like form. There is a consistent principle of organization which maintains the continuity of our body during our life, and it is this that will persevere in the resurrection. For more of this discussion you can read Become like the Angels: Origen's Doctrine of the Soul by Benjamin Blosser.

Origen's attempts to spiritualize the resurrected body are met with great resistance. For example, Methodius of Olympus, living on shortly after Origen, takes great issue with it, saying that Origen's idea of a spiritual body is a denial of the resurrection of the body. It is far more mainstream, especially in Latin writers, to think of the resurrected body as being much like our current bodies, but without the pains and infirmities that are a result of our fallen condition. The Origenist doctrine of the resurrection comes to a head in the late 4th early 5th century Origenist controversy, largely centered around Jerome and Rufinus, where Jerome accuses Rufinus of denying the resurrection because of his Origenism. Much more can be read about this in The Origenist Controversy: The Cultural Construction of an Early Christian Debate by Elizabeth Clark. In the Papal condemnation of Origenism that follows this debate and the 6th century condemnations of Origenism by Justinian, we find a central concern being, besides the pre-existence and apokatastasis of souls, Origen's supposed denial of the resurrected body. While it is debatable whether Origen in fact denies this, later followers, such as Evagrius Ponticus, certainly did. Evagrius, for instance, taught in his Letter to Melania that all souls will be "naked" and flow back into God as rivers into the sea. All of which is to say that some Christians tried to spiritualize the resurrected body, but these efforts were met with serious hostility from orthodox Christianity.

We also see discussion in the early Church as to whether there will be food or drink in the resurrection and whether we will have mouths and digestive tracks for it. Per Jesus in Matthew 22:30 ("At the resurrection people will neither marry nor be given in marriage; they will be like the angels in heaven") there will be no sex in heaven, so will we have genitals? All of these are debated questions, but a fair number of Church Fathers seem to think we will not have mouths, nor intestines, nor genitals, since we will have no use for them. The question of what a body looks like without these doesn't seem to receive too much attention. For more discussion of this type of question, read Fr. Brian Daley, S.J.'s The Hope of the Early Church: A Handbook of Patristic Eschatology.

To sum up, the early Church certainly taught the resurrection of the body, but it wasn't as simple as our bodies being exactly the same as they are now. There were attempts to spiritualize the resurrected body, but these were not met with a great deal of acceptance, though somebody like St. Gregory of Nyssa seems to get away with it somewhat. I hope that helps, and I'm sorry I couldn't offer anything about your second question concerning cremation.