I'd say for sure Katanas are probably "masterwork" versions of bastard swords, with an asian flavour twist.
No, not really - they lack the raw mass to do the job of a Bastard Sword. A bastard sword can kill a man in plate by caving in his chest (it won't cut through his armor). It can also parry heavier weapons in a pinch. The Katana will probably break.
but those three swords are clearly different in terms of real-life damage to both armored and unarmored opponents. The exact proportions are not germane to the discussion of D&D, except insomuch as the granularity of damage die need to account for their relative merits. And right now, they don't. Which is a problem.
Again, not really. Swinging one-handed greatly reduces the effectiveness of those weapons against armor, shields, and other weapons. The bastard sword will still carry a lot of momentum, but it's much slower. The katana loses a huge amount of its strength one-handed, needing to rely on the keenness of its edge vs. a lack of armor on the enemy.
If you want to handle such a two-handed weapon in one hand ask your DM if you can use the next weapon profile down - the scimitar or the long sword. The drop in power (katana) or speed (bastard sword) can generously be worked out as a single demotion of the damage die.
Seriously, you using a bastard sword one-handed if you REALLY need to lunge, hold onto a shield (sub-optimal), hold onto reigns (there are better weapons when mounted by far) or your off-hand is injured and you can't find a lighter weapon.
Likewise you use a katana one-handed only if you have an off-hand weapon (mostly a defensive tactic), you have to strike from a drawing stance (iaido), you are mounted, or your off-hand is injured and you need the superior reach over the wakizashi.
And yes, D&D doesn't differentiate the reach and threat-zone differences between longer and shorter swords either, except imprecisely by the damage die. There's a very good reason why large swords, axes, and hammers fell out of use once heavy armor went obsolete - faster and lighter piercing and cutting weapons (rapier, epee, saber) are much, much more deadly when armor is out of the picture.
- Marty Lund