On the second mission now and I am curious... is there more than one upgrade for his sword? And does it change the finish (color) every time? Thanks in advance.

Nope the sword has 1 upgrade. The Sword Crossing one. There are no blueprints for other upgrades and you don't get anymore as the story goes on.

I am going for "Clean Hands" and so do not use sword at all. So Corvo's Sword stays perfectly clean also.
As there is not much use for money I found, so I usually spend it for weapon/ammo upgrades. And I was a lot frightened after purchasing sword upgrade - it became covered in blood as I was a butcher! I replayed this moment, but it still goes the same. I am still to finish the game so I hope it is just a visual bug.

That sounds like a useful upgrade for me, since I rarely get through a level without having to fight for my head at least once.