Can you transfuse a damaged structure while it is being built?

Well. I've created a video answer to your video question!

Spoiler: NO. You cannot transfuse a building, as the message "Cannot target units under construction" states. What it also means is that you cannot transfuse larva-eggs and (I've not tested it, but think that it's the same) morphing units.

That's a negative: http://us.battle.net/sc2/en/forum/topic/3424682535

Transfuse a building while it is being built is equivalent to terran being able to repair a building while it is being built.

People would be crying foul, saying that Terrans can't repair-build and Protoss can't Chronoboost-build.

As for repairing it if it's damaged while building, I don't think that's possible either (for the same reasons that Terran and Protoss can't).

One poster goes on to mention the balance issues that this caused in WarCraft III, with humans being able to speed build stuff. Blizzard probably wanted to avoid this with SC2.