Order of the White Wing

Does the minion that's being summoned from elsewhere enter the realm for a moment before it's banished?

No, it doesn't ever enter the realm. For example, it will not trigger a Genesis, it will not get a Lance, it will not trigger a Mariner's Curse, etc. It doesn't actually enter the realm at all, since this is a replacement effect.

Does this prevent minions from being cast from other 'hand' zones (e.g. Morgana le Fay, or an Omphalos)?

Yes. The Order prevents minions from being summoned from anywhere except directly from a player's hand.