Not bad, I would like to add more:
As I take hold if the chunk of ice I close my eyes and feel the ice I feel that it is rigid and blunt, but the shape of it signifies that it is a versatile element that can be used for many purposes offensive and defensive alike. It also has a gold feeling to it (

) that makes you feel weak to it, yet I feel attracted to it and it's power it's uniqueness and "character".
Yes ice is made up of water and wind chakra elements, the user will freeze the water by applying wind chakra to it and rotating his chakra around it until the water freezes. This feet can't be accomplished if the user is not a member of the Yuki clan, nevertheless others from the movies were capable of manipulating snow, this for them was a hidden ability much like the Naras(Cries about missing my Nara days :T_T

. Those were incapable of creating ice only manipulating it.
Now onto the strengths, weaknesses, and neutrality of the element:*
Ice is as you said double strong against water, meaning a S rank water technique can be stopped by a B rank ice technique, but you must also take into consideration size and mass. If an S rank water technique that covers extensive range goes up against the B rank version of needles of death it won't stop it since the needles are to small and weight much less than the water technique. However it can also be used to stop the water of used correctly. We will get to that eventually.
Now for ice and wind, also like you said wind is weak against ice, not much to add on this topic unless you have some questions regarding it.
Now to weaknesses, Ice is like you said weak to lightning, simply because the lightning will pierce though the ice and shatter it. Now for earth you are mistaken Ice is not weak to all earth techniques, but only to penetrating earth based techniques such as jagged pillars or weapons of earth.
And finally to neutrality, Ice is neutral to fire as when a fire and ice technique of same rank collide the results will be as the following: the fire would overpower the wind in the ice technique, but as soon as that happens the water from the technique will cause the fire to get doused.
Also this is for you mister wood man, ice and wood are neutral against each other.
The last thing I would like to 'say' is that ice is indeed versatile like wood, and IMO even more than wood. In ice you can create disks that can help you fly, you can cause blinding through specific techniques, and you can even burn, yes I said burn, you can even burn/create fires with ice. And all the above were in one technique

.
Also shields/armors/pillars/weapons/icicles/monsters/animals can be created though ice release.
Any Questions?