The default name for your daughter is Olive Oyl; she's named after the character in the Popeye series. You can get the default daughter by typing a bunch of random letters and symbols when prompted for a name at the beginning.
This default name exists for both the Japanese version and the beta English version.
Yes, the "Gainax" developer is the famous animation studio we all know and love, they make lots computer software (based on both their own work and that of some manga artists) and the Princess Maker series is one of its most famous ones. In fact the character designer for the whole series is none other than Takami Akai, long-time member of Gainax and assistant director on 1987's Royal Space Force: The Wings of Honneamise.
The English translation, being a beta, has a special debug mode which allows you to cheat.