The name of this style of knife

Absolutely not. A honing steel is significantly harder than the blade of a knife and is specially textured for honing.

Rubbing two knives together is more likely to dull or knick the blades than anything else.

The blade of a knife should never touch anything harder than a wooden cutting board. Not glass, not granite, and certainly not steel (except for honing/sharpening).

Both the Santoku and French knives will work for the same types of things, so a lot of it comes down to preference. Santoku knives are lighter, so this can lead to less hand strain and quicker cutting. One thing that the Santoku are very good at is very thin slicing of vegetables, for two reasons: first, as you point out, you do not use a rocking motion, but rather chop down in one motion, which with practice can be quicker and more efficient. Second, Santoku knives usually have a much thinner blade angle (around 15 degrees vs 30-40 degrees on a French knife). This is because one side of a Santoku is flat, and the other side is beveled (like a chisel), so you only sharpen one side. To accommodate this thin blade the Santoku knives will be made with a harder steel, which helps maintain blade sharpness, but may increase the propensity for chipping if misused, and also makes them harder to sharpen.

Still, personal preference dictates which knife to use. Many people prefer the Gyuto style of knife, which (roughly speaking) combines features of both Santoku knifes and French knives. They are made of a hardened steel, are sharpened on both sides, but maintain edges around 22-26 degrees. The Gyutos, like the Santokus are fairly light. Gyutos also have a rounded belly but it is less pronounced than on a French knife.

There is much more to be said, but basically the thing to do is get your hands on some knives and start experimenting--see what you like!