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 feat 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). Those were incapable of creating ice only manipulating it.
Now onto the strengths, weaknesses, and neutrality of the element:
Ice is double strong against water not only strong like you said, 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.
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 also 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, for others that are blunt they are neutral.
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 ice and wood are neutral to each other..
The last thing I would like to say is that ice is versatile like wood release, 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?