You're pretty much spot on with everything, though I had hoped you'd mention Temari who uses her fan to augment her Wind jutsu instead of using hand seals, but it's no big deal. I'll help you with the other elements.
Wind when used with Water is used to help push the water. When you do this, you're creating a stronger pushing force with the secondary element. Though it wouldn't increase the rank or anything, it's better if you're using something fast in which you can make it even faster, leaving less room for them to come up with a counter.
When you combine it with Earth, it's more or less the same effect with Water, but you can be a little more versatile with it. Seeing as Earth has more than a solid release and you can create other things like mud, dust clouds and so on, Wind can help propagate that.
So let's talk about counters. What would happen if a Wind technique coming from you and a Fire technique coming from me both of the same rank collide with each other?