So... About Oscar:
Does anyone else want to see him get seriously ticked off by Qrow calling him 'Oz' and insisting that he is his own person, not just a reincarnation of Professor Ozpin? It might be interesting as a character moment for Qrow too; the realisation that he's going to have to stay the leader of the Resistance because the current holder of the Vale Artefact is a kid Ruby's age who hears Oz giving him advice.
It's also a great character moment for Oscar himself the way i see it. No matter who is living in his head, he still wants to be his own person, not just a way for qrow and others to connect to Ozpin through. Because if the others see him as an individual other than Oz, he'll be able to grow more and do more alongsides the others.
Although Oz could easily be short for Oscar too, so... i think it's more about how they treat him than what they call him :)