In an infant, the only "sin" involved is the effect of the ancestral sin.
In someone of the age of reason, the baptism is for the removing of the effects of the ancestral sin *and* for the remission of sins committed by the person. Hence Ananias and his exhortation to St. Paul to "be baptized and wash away thy sins."
I'm repeating what I have been told. I'm not sure what the written guidelines of the Church say about all of this. It could be that I was being told something that is not traditional Orthodox practice.
I again find it unlikely that the Orthodox practice of a relatively late baptism would have developed had the parents known that their child would go without an Orthodox burial in the event of untimely death (which, up until very recent times, was a far from uncommon occurence.)
But maybe not. Regardless of what happens in funeral practices, I don't think that this would change Orthodox beliefs that all infants go to heaven -- which seems to be virtually universal in my own experience. Maybe I've just been around softies all these years.
Maybe. Perhaps the priest would conduct the service and leave out some litanies and hymns that would not be appropriate, those specifically applicable to Orthodox Christians.
As you know, the Orthodox Church can and does pray for all, and not just for Orthodox Christians, and this goes for the dead as well, whether they are baptized or not.
But that is not the same as the funeral for the Orthodox Christians, which is intended only for those who have been baptized Orthodox.
As for the late baptism, that is the parents' decision and I don't believe the Church teaches that one should not baptize them as early as possible. I specifically recall bishop Eusebius (yes, the Church historian) answering questions regarding this in relation to the 8th day, as in Judaism, and his reply that it should be done as soon as possible.