Does the density of ice increase further when it is further cooled? As we know, water becomes ice at $0^°$C.
After becoming ice, if we lower the temperature further ($<0^°$C), then does the density of ice increase further?
If yes, in that case, does the volume decrease?
 A: From Wikipedia: Ice § Physical properties, we learn that the density of liquid water at 0 °C is about 1.00 g/cm³, and the density of ice at 0 °C is about 0.917 g/cm³. So at the transition from liquid water at 0 °C to ice also at 0 °C, we have a decrease in density corresponding to an expansion of the substance.
You ask what happens to that ice if the temperature is lowered. The same source writes:

Density of ice increases slightly with decreasing temperature
and has a value of 0.9340 g/cm³ at −180 °C (93 K).

So if you cool further, the density increases corresponding to a contraction of the substance.
You:

If yes, in that case, does the volume decrease?

Yes, exactly, the volume of ice decreases when the temperature decreases!
We present a graph (source) that shows both phases:

A: 
After becoming ice, if we lower the temperature further($<0^°$C), then does the density of ice increase further? If yes, in that case, does the volume decrease?

From Density Anomalies of Water
, we learn that the density variation of water with temperature varies in the following way:

Therefore, the density of water will stray away from its maximum density as it is cooled. The temperature below $0^°$C at which it will be the closest to the maximum density (at $4^o$C) is, well, $0^°$C because the packing of the molecules will not become any tighter. As can be seen, as you lower the temperature in the ice region, the density of water get increases which leads to the contraction of ice (Note there is a discontinuity at the point of phase change).  For the volume which is inversely proportional to density, things go another way around. For a detailed discussion of the peculiar nature of water see here,here
Edit :
A better picture which shows the discontinuity during the phase transition:

More on this here.
