You can use iaido with a straight blade. It's not as efficient, but it's doable. One or two edges doesn't matter, though. But as you say, having the blade drawn beforehand is always better, unless you're aiming for the surprise. In any combat situation, you have your sword drawn unless you're using another weapon.
However, a katana is not a saber. A saber is a specific type of curved backsword, not any single-edged sword. They're similar in many aspects, but as far as I know a saber is never a two-handed sword, while a katana is (though it can be used in one hand, similar to a longsword).
The swords they use here look like some generic fantasy version of early arming swords.
On a tangent, in Japanese "katana" is often used colloquially to mean just "sword" in general, but the English meaning is that specific type of Japanese sword. In most of the world, for their time periods, most swords were just called "swords" in their local language, and the designations we use for them are almost all modern inventions.
Yeah, it is for drawing and attacking quickly. While learning it you might also learn other sword techniques, but I'm not too familiar about that, and it would depend on the school. It's also a common misconception and portrayal in Japanese media that iaido is faster than just attacking with an already drawn blade, which doesn't make sense, since you're adding a restriction.